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Students will be able to write mathematical proofs and reason abstractly in exploring properties of groups and ringsUse the division algorithm, Euclidean algorithm, and modular arithmetic in computations and proofs about the integersDefine, construct examples of, and explore properties of groups, including symmetry groups, permutation groups and cyclic groupsDetermine subgroups and factor groups of finite groups, determine, use and apply homomorphisms between groupsDefine and construct examples of rings, including integral domains and polynomial rings.
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1
MTH376: Algebra
For
Master of Mathematics
By
Dr. M. Fazeel AnwarAssistant Professor
Department of Mathematics, CIIT Islamabad
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Lecture 02
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Recap
• Set• Some important number sets• Function
A function is a rule that assigns to each element in a unique element in . Mathematically is a function if
i. for all
ii. implies .
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Binary Operation
• A binary operation on a set is a rule which assigns to every ordered pair of elements of an element is
• More precisely is a binary operation if
is a function.
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Examples
• Let denotes the operation of addition on the set of integers Then is a binary operation. Similarly addition is a binary operation on and
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Examples(continued…)
• Let denotes the operation of multiplication on the set of integers Then is a binary operation. Similarly multiplication is a binary operation on and
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More examples
• Let be the set of real valued functions defined for all real numbers, then the usual sum and product of functions are binary operations on
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Some non examples
• Subtraction is not a binary operation on Also division is not a binary operation of .
• The operation defined by is not a binary operation. Also is not a binary operation.
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Motivation for defining groups
Solution of linear equations
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Motivation (continued…)
Solve the following equation:
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Group
• A group is a set together with a binary operation on such that the following axioms are satisfied:
1. The binary operation is associative.
2. There is an element such that for all
3. For each there is an element such that for all
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Discussion and remarks
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Summary
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Thank You