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Math 3121Abstract Algebra I
Lecture 3Sections 2-4 Binary Operations
Definition of Group
Questions on HW (not to be handed in)
bull Pages 19-20 1 3 5 13 17 23 38 41
Section 2 Binary Relations
bull Definition A binary relation on a set S is a function mapping StimesS into S For each (a b) in StimesS denote ((a b)) by a b
Examples
bull The usual addition forℤ the integersℚ the rational numbersℝ the real numbersℂ the complex numbersℤ+ the positive integersℚ + the positive rational numbersℝ + the positive real numbersbull The usual multiplication for these numbers
Counterexamples
bull If is a binary operation on a S then ab must be defined for all a b in S
bull Examples of where it is not bull subtraction on ℝ +bull division on Zbull More
More examples
bull Matricesndash Addition is a binary operation on real n by m
matricesndash What about multiplication
bull Functionsndash For real valued functions of a real variable
addition multiplication subtraction and composition are all binary operations
Closure
bull Suppose is a binary operation on a set S and H is a subset of S The subset H is closed under iff ab is in H for all a b in H In that case the binary operation on H given by restricting to members of H is called the induced operation of on H
bull Examples in book squares under addition and multiplication of positive integers
Commutative and Associative
bull Definition A binary operation on a set S is commutative iff for all a b in S
a b = b abull Definition A binary operation on a set S is
associative iff for all a b c in S(a b) c = a (b c)
Examples
bull Composition is associative but not commutative
bull Matrix multiplication is associative but not commutative
bull More in book
Section 3
bull Definition of binary structurebull Homomorphismbull Isomorphismbull Structural propertiesbull Identity elements
Binary Structures
bull Definition (Binary algebraic structure) A binary algebraic structure is a set together with a binary operation on it This is denoted by an ordered pair lt S gt in which S is a set and is a binary operation on S
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Questions on HW (not to be handed in)
bull Pages 19-20 1 3 5 13 17 23 38 41
Section 2 Binary Relations
bull Definition A binary relation on a set S is a function mapping StimesS into S For each (a b) in StimesS denote ((a b)) by a b
Examples
bull The usual addition forℤ the integersℚ the rational numbersℝ the real numbersℂ the complex numbersℤ+ the positive integersℚ + the positive rational numbersℝ + the positive real numbersbull The usual multiplication for these numbers
Counterexamples
bull If is a binary operation on a S then ab must be defined for all a b in S
bull Examples of where it is not bull subtraction on ℝ +bull division on Zbull More
More examples
bull Matricesndash Addition is a binary operation on real n by m
matricesndash What about multiplication
bull Functionsndash For real valued functions of a real variable
addition multiplication subtraction and composition are all binary operations
Closure
bull Suppose is a binary operation on a set S and H is a subset of S The subset H is closed under iff ab is in H for all a b in H In that case the binary operation on H given by restricting to members of H is called the induced operation of on H
bull Examples in book squares under addition and multiplication of positive integers
Commutative and Associative
bull Definition A binary operation on a set S is commutative iff for all a b in S
a b = b abull Definition A binary operation on a set S is
associative iff for all a b c in S(a b) c = a (b c)
Examples
bull Composition is associative but not commutative
bull Matrix multiplication is associative but not commutative
bull More in book
Section 3
bull Definition of binary structurebull Homomorphismbull Isomorphismbull Structural propertiesbull Identity elements
Binary Structures
bull Definition (Binary algebraic structure) A binary algebraic structure is a set together with a binary operation on it This is denoted by an ordered pair lt S gt in which S is a set and is a binary operation on S
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Section 2 Binary Relations
bull Definition A binary relation on a set S is a function mapping StimesS into S For each (a b) in StimesS denote ((a b)) by a b
Examples
bull The usual addition forℤ the integersℚ the rational numbersℝ the real numbersℂ the complex numbersℤ+ the positive integersℚ + the positive rational numbersℝ + the positive real numbersbull The usual multiplication for these numbers
Counterexamples
bull If is a binary operation on a S then ab must be defined for all a b in S
bull Examples of where it is not bull subtraction on ℝ +bull division on Zbull More
More examples
bull Matricesndash Addition is a binary operation on real n by m
matricesndash What about multiplication
bull Functionsndash For real valued functions of a real variable
addition multiplication subtraction and composition are all binary operations
Closure
bull Suppose is a binary operation on a set S and H is a subset of S The subset H is closed under iff ab is in H for all a b in H In that case the binary operation on H given by restricting to members of H is called the induced operation of on H
bull Examples in book squares under addition and multiplication of positive integers
Commutative and Associative
bull Definition A binary operation on a set S is commutative iff for all a b in S
a b = b abull Definition A binary operation on a set S is
associative iff for all a b c in S(a b) c = a (b c)
Examples
bull Composition is associative but not commutative
bull Matrix multiplication is associative but not commutative
bull More in book
Section 3
bull Definition of binary structurebull Homomorphismbull Isomorphismbull Structural propertiesbull Identity elements
Binary Structures
bull Definition (Binary algebraic structure) A binary algebraic structure is a set together with a binary operation on it This is denoted by an ordered pair lt S gt in which S is a set and is a binary operation on S
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Examples
bull The usual addition forℤ the integersℚ the rational numbersℝ the real numbersℂ the complex numbersℤ+ the positive integersℚ + the positive rational numbersℝ + the positive real numbersbull The usual multiplication for these numbers
Counterexamples
bull If is a binary operation on a S then ab must be defined for all a b in S
bull Examples of where it is not bull subtraction on ℝ +bull division on Zbull More
More examples
bull Matricesndash Addition is a binary operation on real n by m
matricesndash What about multiplication
bull Functionsndash For real valued functions of a real variable
addition multiplication subtraction and composition are all binary operations
Closure
bull Suppose is a binary operation on a set S and H is a subset of S The subset H is closed under iff ab is in H for all a b in H In that case the binary operation on H given by restricting to members of H is called the induced operation of on H
bull Examples in book squares under addition and multiplication of positive integers
Commutative and Associative
bull Definition A binary operation on a set S is commutative iff for all a b in S
a b = b abull Definition A binary operation on a set S is
associative iff for all a b c in S(a b) c = a (b c)
Examples
bull Composition is associative but not commutative
bull Matrix multiplication is associative but not commutative
bull More in book
Section 3
bull Definition of binary structurebull Homomorphismbull Isomorphismbull Structural propertiesbull Identity elements
Binary Structures
bull Definition (Binary algebraic structure) A binary algebraic structure is a set together with a binary operation on it This is denoted by an ordered pair lt S gt in which S is a set and is a binary operation on S
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Counterexamples
bull If is a binary operation on a S then ab must be defined for all a b in S
bull Examples of where it is not bull subtraction on ℝ +bull division on Zbull More
More examples
bull Matricesndash Addition is a binary operation on real n by m
matricesndash What about multiplication
bull Functionsndash For real valued functions of a real variable
addition multiplication subtraction and composition are all binary operations
Closure
bull Suppose is a binary operation on a set S and H is a subset of S The subset H is closed under iff ab is in H for all a b in H In that case the binary operation on H given by restricting to members of H is called the induced operation of on H
bull Examples in book squares under addition and multiplication of positive integers
Commutative and Associative
bull Definition A binary operation on a set S is commutative iff for all a b in S
a b = b abull Definition A binary operation on a set S is
associative iff for all a b c in S(a b) c = a (b c)
Examples
bull Composition is associative but not commutative
bull Matrix multiplication is associative but not commutative
bull More in book
Section 3
bull Definition of binary structurebull Homomorphismbull Isomorphismbull Structural propertiesbull Identity elements
Binary Structures
bull Definition (Binary algebraic structure) A binary algebraic structure is a set together with a binary operation on it This is denoted by an ordered pair lt S gt in which S is a set and is a binary operation on S
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
More examples
bull Matricesndash Addition is a binary operation on real n by m
matricesndash What about multiplication
bull Functionsndash For real valued functions of a real variable
addition multiplication subtraction and composition are all binary operations
Closure
bull Suppose is a binary operation on a set S and H is a subset of S The subset H is closed under iff ab is in H for all a b in H In that case the binary operation on H given by restricting to members of H is called the induced operation of on H
bull Examples in book squares under addition and multiplication of positive integers
Commutative and Associative
bull Definition A binary operation on a set S is commutative iff for all a b in S
a b = b abull Definition A binary operation on a set S is
associative iff for all a b c in S(a b) c = a (b c)
Examples
bull Composition is associative but not commutative
bull Matrix multiplication is associative but not commutative
bull More in book
Section 3
bull Definition of binary structurebull Homomorphismbull Isomorphismbull Structural propertiesbull Identity elements
Binary Structures
bull Definition (Binary algebraic structure) A binary algebraic structure is a set together with a binary operation on it This is denoted by an ordered pair lt S gt in which S is a set and is a binary operation on S
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Closure
bull Suppose is a binary operation on a set S and H is a subset of S The subset H is closed under iff ab is in H for all a b in H In that case the binary operation on H given by restricting to members of H is called the induced operation of on H
bull Examples in book squares under addition and multiplication of positive integers
Commutative and Associative
bull Definition A binary operation on a set S is commutative iff for all a b in S
a b = b abull Definition A binary operation on a set S is
associative iff for all a b c in S(a b) c = a (b c)
Examples
bull Composition is associative but not commutative
bull Matrix multiplication is associative but not commutative
bull More in book
Section 3
bull Definition of binary structurebull Homomorphismbull Isomorphismbull Structural propertiesbull Identity elements
Binary Structures
bull Definition (Binary algebraic structure) A binary algebraic structure is a set together with a binary operation on it This is denoted by an ordered pair lt S gt in which S is a set and is a binary operation on S
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Commutative and Associative
bull Definition A binary operation on a set S is commutative iff for all a b in S
a b = b abull Definition A binary operation on a set S is
associative iff for all a b c in S(a b) c = a (b c)
Examples
bull Composition is associative but not commutative
bull Matrix multiplication is associative but not commutative
bull More in book
Section 3
bull Definition of binary structurebull Homomorphismbull Isomorphismbull Structural propertiesbull Identity elements
Binary Structures
bull Definition (Binary algebraic structure) A binary algebraic structure is a set together with a binary operation on it This is denoted by an ordered pair lt S gt in which S is a set and is a binary operation on S
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Examples
bull Composition is associative but not commutative
bull Matrix multiplication is associative but not commutative
bull More in book
Section 3
bull Definition of binary structurebull Homomorphismbull Isomorphismbull Structural propertiesbull Identity elements
Binary Structures
bull Definition (Binary algebraic structure) A binary algebraic structure is a set together with a binary operation on it This is denoted by an ordered pair lt S gt in which S is a set and is a binary operation on S
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Section 3
bull Definition of binary structurebull Homomorphismbull Isomorphismbull Structural propertiesbull Identity elements
Binary Structures
bull Definition (Binary algebraic structure) A binary algebraic structure is a set together with a binary operation on it This is denoted by an ordered pair lt S gt in which S is a set and is a binary operation on S
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Binary Structures
bull Definition (Binary algebraic structure) A binary algebraic structure is a set together with a binary operation on it This is denoted by an ordered pair lt S gt in which S is a set and is a binary operation on S
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Homomorphism
bull Definition (homomorphism of binary structures) Let ltSgt and ltSrsquorsquogt be binary structures A homomorphism from ltSgt to ltSrsquorsquogt is a map h S Srsquo that satisfies for all x y in S
h(xy) = h(x)rsquoh(y)bull We can denote it by h ltSgt ltSrsquorsquogt
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Examples
bull Let f(x) = ex Then f is a homomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is a homomorphism from the real numbers with addition to unit circle in the complex plane
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Isomorphism
bull Definition A homomorphism of binary structures is called an isomorphism iff the corresponding map of sets is one to one and onto
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Examples
bull Let f(x) = ex Then f is an isomorphism from the real numbers with addition to the positive real numbers with multiplication
bull Let g(x) = eix Then g is not an isomorphism from the real numbers with addition to unit circle in the complex plane (not 1-1)
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Identity Element
bull Definition Let ltS gt be a binary structure An element e of S is an identity element for iff for all x in S
e x = x e = x
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Uniqueness of Identity
Theorem A binary structure has at most one identity element
Proof Let ltS gt be a binary structure If e1 and e2 are both identities thene1 e2 = e1 because e2 is an identity and
e1 e2 = e2 because e1 is an identity
Thus e1 = e2
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Preservation of identity
bull Theorem If h ltS gt ltSrsquorsquogt is an isomorphism of binary algebraic structures and e is an identity element of ltS gt then h(e) is an identity element of ltSrsquorsquogt
bull Proof Let e be the identity element of S For each xrsquo in Srsquo There is an x in S such that h(x) = xrsquo Then h(x e) = h (e x) = h(x) By the homomorphism property h(x) h(e) = h (e) h(x) = h(x) Thus xrsquo h(e) = h(e) xrsquo = xrsquo Thus h(e) is the identity of Srsquo
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Section 4 Groups
bull Definition A group ltG gt is a set G together with a binary operation on G such that1) Associatively For all a b c in G
(a b) c = a (b c)
2) Identity There is an element e in G such that for all x in G
e x = x e = x
3) Inverse For each x in G there is an element xrsquo in G such that
x xrsquo = xrsquo x = e
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Technicalities
bull Sometimes use notation ltG e lsquogt to denote all its components
bull Specifying the inverse of each element x of G is a unary operation on the set G
bull Specifying the identity call be considered an operation with no arguments (n-ary where n = zero)
bull We often drop the ltgt notation and use the set to denote the group when the binary operation is understood We already drop the e and lsquo Later we will show they are uniquely determined
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Examplesbull ltℤ +gt the integers with additionbull ltℚ +gt the rational numbers with additionbull ltℝ +gt the real numbers with additionbull ltℂ +gt the complex numbers with additionbull The set -1 1 under multiplicationbull The unit circle ltU gt in the complex plane under
multiplicationbull Many more in the book bull positive rationals and reals under multiplicationbull nonzero rationals and reals under multiplicationbull N by M real matrices under addition
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Elementary Properties of Groups
bull Cancellation Left ab = ac implies b = cRight ba = ca implies b = c
bull Unique solutions of ax = bbull Only one identitybull Formula for inverse of product
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Cancellation
Theorem If G is a group with binary operation then left and right cancellation hold
a b = a c imply b = cb a = c a imply b = c
Proof Suppose a b = a c Then there is an inverse arsquo to a Apply this inverse on the left
arsquo (a b) = arsquo (a c)(arsquo a ) b = (arsquo a) c associativelye b = e c inverseb = c identity
Similarly for right cancellation
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
First Order Equations
Theorem Let ltG gt be a group If a and b are in G then ax = b has a unique solution and so does xa = b
Proof (in class ndash solve for x by applying inverse uniqueness follows by cancellation)
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Uniqueness of Identity and Inverse
bull Theorem Let ltG ersquogt be a group 1) There is only one element y in G such that y x = x y = x for all x in G and that element is e 2) For each x in G there is only one element y such that yx = xy = e and that element is xrsquo
bull Proof 1) is already true for binary algebraic structures 2) proof in class (use cancellation)
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
Formula for inverse of product
Theorem Let ltG e lsquogt be a group For all a b in G the inverse is given by (a b)rsquo = brsquoarsquo
Proof Show it gives and inverse (in class)
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
HW (due Tues Oct 7)
bull Not to hand in Pages 45-49 1 3 5 21 25bull Hand in pages 45-49 2 19 24 31 35
