ASSIGNMENT BOOKLET

Bachelor’s Degree Programme

Abstract Algebra (MTE-06)

(Valid from 1st January, 2013 to 31st December, 2013)

School of Sciences

Indira Gandhi National Open University

Maidan Garhi

New Delhi-110068

(For January, 2013 Cycle)

MTE-06

It is compulsory to submit the assignment before filling in the exam form.

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Dear Student,

Please read the section on assignments in the Programme Guide for Elective Courses that we sent you

after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous

evaluation, which would consist of a tutor marked assignment for this course. The assignment is in this

booklet.

Instructions for Formating Your Assignments

Before attempting the assignment please read the following instructions carefully.

1) On top of the first page of your answer sheet, please write the details exactly in the following format :

ROLL NO:………………………….

NAME:………………………….

ADDRESS:………………………….

…………………………..

…………………………..

…………………………..

COURSE CODE:………………………….

COURSE TITLE:………………………….

ASSIGNMENT NO:………………………

STUDY CENTRE:………………………...

DATE:…………………….……………….

PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND

TO AVOID DELAY.

2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers.

3) Leave a 4 cm. margin on the left, top and bottom of your answer sheet.

4) Your answers should be precise.

5) While solving problems, clearly indicate which part of which question is being solved.

6) This assignment is valid only upto December, 2013. If you fail in this assignment or fail to

submit it by December, 2013, then you need to get the assignment for January, 2014 session

and submit it as per the instructions given in the programme guide.

7) It is compulsory to submit the assignments before filling the examination form.

Please retain a copy of your answer sheets.

Wish you luck!

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ASSIGNMENT (To be done after studying the Blocks.)

Course Code : MTE-06

Assignment Code : MTE-06/TMA/2013

Total Marks : 100

1) Which of the following statements are true ? Give reasons for your answers.

i) For any set S, S S is an equivalence relation.

ii) If H G such that .GHthen,3H:G

iii) The group of units of ,13Z

Z with respect to multiplication, is cyclic.

iv) 20Z is an abelian group with no proper non-trivial normal subgroups (i.e., it is simple).

v) A group of order 168 has either 1 or 8 elements of order 7. vi) In Z100 there is a unique zero divisor which is also a unit.

vii)

Rc,b,a

0c

bais a ring with respect to the usual matrix addition and

multiplication. viii) Every subring of a commutative ring R is an ideal of R. ix) Any abelian group is a ring with respect to a suitably defined multiplication.

x) If I and J are ideals in a ring R, then IJ .JI (20)

2a) An isometry of R2 is a map 22: RR which preserves the Euclidean distance between any

two points in R2, i.e., y,xyx)y()x( R

2.

It is known that if ),0,0()0,0( then is a linear invertible map, and is called a linear

isometry.

Prove that the set L of linear isometries of R2 is a group with respect to the composition of

functions. (5)

b) Let (G, ) be a group with 5 elements. Construct a Cayley table for . How many distinct

Cayley tables can you construct ? (5)

3a) Use the Fundamental Theorem of Homomorphism to prove that 12

Z ~ g iff g is an element

of order 12 in a group (G, ). (7)

b) Obtain two distinct elements of ,7Z

Z and two distinct subgroups of Z

Z7

. (3)

4a) If )G(Z

G is cyclic, then show that G is abelian. (3)

b) Consider (Z, +) (Q, +).

i) Show that Z4

7 has order 4 in .

ZQ

ii) Find o ,b

a

Z where (a,b) = 1, b .0

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iii) Show that every element in Z

Q has finite order, and that there are elements in Z

Q of

order n for any .n N

iv) Show that Z

Q is an infinite group that is not cyclic. (10)

c) Prove that Δ S4, where ).4321( (4)

d) Give an example, with justification, of a ring R which is not a domain, but for which every ideal

is principal. (3)

5a) Prove or disprove that we can define a multiplication on R3 = R c,b,a|cjbia such that

.j1i 22 (5)

b) Use the Euclidean algorithm to find a g.c.d. of 1xxxand1xx 345 in Z2[x]. (3)

c) If SR:f is a ring homomorphism, then show that char R char S. (2)

6a) Prove that a ring with characteristic zero contains a subring isomorphic to Z. (5)

b) Give an example, with justification, of a ring R which is not a domain, but for which every

deal is principal. (3)

c) Give an example, with justification, of a UFD whose quotient field is C. (2)

7a) Use the Fundamental Theorem of Homomorphism to prove that 2

5

ZZ Z

3. (6)

b) Check whether the following are ring homomorphisms.

i) mm z)z(f::f ZZ

ii)

pp

00)p(f:)(:f 2 QMQ

iii) ,A\X)A(f:)X(P),),X(P(:f where P(X) is the power set of the set X.

Further, for those f that are ring homomorphisms, what does the Fundamental Theorem of

Homomorphism give ? (9)

c) Let Z[w] = Z y,x|wyx , where w is a cube root of unity.

Prove that

i) N(z) = z .wyxzwhere,yxyxz 22

ii) [w].zz),z(N)z(N)zz(N 212121 Z

iii) [w]z Z is a unit iff N(z) = 1. (5)