Download MTE-06 Abstract Algebra

MTE-06
Assignment Booklet
Bachelor’s Degree Programme
Abstract Algebra
School of Sciences
Indira Gandhi National Open University
New Delhi
2006
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 two tutor-marked assignments for this course. Both these
assignments are 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) The assignment responses are to be submitted to your Study Centre Coordinator by November, 2006,
not later.
Please retain a copy of your answer sheets.
Wish you luck!
2
ASSIGNMENT 1
(To be done after studying Blocks 1 and 2.)
Course Code : MTE-06
Assignment Code : MTE-06/TMA-1/2006
Total Marks : 100
1)
Which of the following statements are true ? Give reasons for your answers.
i)
ii)
iii)
Every semigroup is not a group.
For any set S, S  S is an equivalence relation.
If H  G such that G : H  3, then H  G.
iv)
Inn G = nZ, if G = Z
v)
If H  G such that G H is cyclic, then G is abelian.
, with respect to multiplication, is cyclic.
The group of units of Z
13Z
Z 20 is an abelian group with no proper non-trivial normal subgroups (i.e., it is simple).
Z  (Z \ N)  {IGNOU } is a set.
A group of order 168 has either 1 or 8 elements of order 7.
(20)
 : N  N  N : x  y  gcd(x, y) is associative.
vi)
vii)
viii)
ix)
x)
2a)
nZ
.
An isometry of R2 is a map  : R 2  R 2 which preserves the Euclidean distance between any
two points in R2, i.e., (x)  ( y)  x  y  x, y  R2.
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)
3a)
b)
4a)
b)
Let (G, ) be a group with 5 elements. Construct a Cayley table for  . How many distinct
Cayley tables can you construct ?
(5)
Use the Fundamental Theorem of Homomorphism to prove that Z
12
~ g  iff g is an element
of order 12 in a group (G,  ).
(7)
Obtain two distinct elements of Z 7Z , and two distinct subgroups of Z 7 Z .
(3)
If G Z(G ) is cyclic, then show that G is abelian.
(3)
Consider (Z, +)  (Q, +).
i)
ii)
iii)
7
 Z has order 4 in Q .
Z
4
a

Find o   Z , where (a,b) = 1, b  0.
b

Show that every element in Q has finite order, and that there are elements in Q of
Z
Z
order n for any n  N.
Show that
3
iv)
Show that Q
Z
is an infinite group that is not cyclic.
(10)
c)
Prove that  Δ S4, where   (1 2 3 4).
d)
Let G be an abelian group containing elements a and b of orders m and n, respectively. Show
that G contains an element whose order is the least common multiple of m and n.
(3)
5a)
b)
c)
6a)
b)
7a)
Prove that Z
143 Z
(4)
is cyclic.
(5)
Find Z(Q8), the center of the group of quaternions. Also, check whether Q 8
Z(Q 8 )
is abelian or
not.
(3)
How many elements of order 7 will a simple group of order 168 have ?
(2)
Let G and H be groups, and  : H  Aut G be a homomorphism. Define a binary operation 
on G  H by  (g, h ), (g , h )   g [ (h ) (g )], h h .
Show that (G  H, ) is a group.
(In fact, (G  H, ) is called the semidirect product of G and H w.r.t. .) .
(5)
Show that a group is abelian iff  : G  G : ( x )  x 1 is an automorphism.
(5)
Prove that the order of a permutation is the least common multiple of the orders of its disjoint
cycles.
(5)
b)
Write (3 7 9) (1 2)  S 9 in 2-line notation. Also obtain its signature.
c)
Give an example, with justification, of a group G with normal subgroups H and K such that
G ~ K, G ~ H but G ~ H  K.
(7)
H
K
8)
Find the Sylow 2- and Sylow 3-subgroups of S4.
4
(3)
(5)
ASSIGNMENT 2
(To be done after studying Blocks 3 and 4.)
Course Code : MTE-06
Assignment Code : MTE-06/TMA-2/2006
Total Marks : 100
1)
Which of the following statements are true ? Give reasons for your answers.
i)
ii)
iii)
iv)
v)
vi)
vii)
viii)
ix)
x)
2a)
In Z100 there is a unique zero divisor which is also a unit.
 a b 

a , b, c  R  is a ring with respect to the usual matrix addition and


 c 0 

multiplication.
Every subring of a commutative ring R is an ideal of R.
Any abelian group is a ring with respect to a suitably defined multiplication.
R  R is a field with respect to the usual operations of addition and multiplication.
If I and J are ideals in a ring R, then IJ  I  J.
A polynomial in R[x] of degree n has at most n roots, where R is a domain.
The characteristic of an infinite field is zero.
The set of all mappings from Z to Z has zero divisors.
If R I is commutative, then so is R, where R is a ring and I is an ideal of R.
(20)
Prove or disprove that we can define a multiplication on R3 = a  bi  cj| a , b, c  Rsuch that
i 2  1  j 2 .
b)
3a)
(5)
Prove that every ring can be embedded in a ring S with identity and characteristic zero.
(10)
Let R = Z[i], n  Z \ {0}, and I = Rn. Then show that a  ib  I iff n|a and n|b. Further, show that
R is a finite ring.
(6)
I
b)
List the ideals in Z 24 Z .
(4)
c)
Prove that a ring with characteristic zero contains a subring isomorphic to Z.
(5)
4a)
b)
c)
5a)
Give an example, with justification, of a ring R which is not a domain, but for which every ideal
is principal.
(3)
Let Z[w] = x  wy | x , y  Z, where w is a cube root of unity.
Prove that
i)
N(z) = z z  x 2  xy  y 2 , where z  x  wy.
N(z1 z 2 )  N(z1 ) N(z 2 ), z1 z 2  Z[w].
ii)
iii)
z  Z[w] is a unit iff N(z) = 1.
iv)
z is irreducible if N(z) is a prime number.
v)
Z[w] is a Euclidean domain.
1  w is a prime element in Z[w].
vi)
(15)
Give an example, with justification, of a UFD whose quotient field is C.
(2)
Use the Euclidean algorithm to find a g.c.d. of x 5  x  1 and x 4  x 3  x  1 in Z2[x].
(3)
5
b)
c)
6a)
b)
List the monic irreducible polynomials of degree 3 in Z3[x].
(3)
Check whether R = Q[x]
is a field or not. Further, if R is a field, obtain its
3x 5  5x 4  5
characteristic. Otherwise, give a field of characteristic 7.
(4)
5
Use the Fundamental Theorem of Homomorphism to prove that Z
Z2
 Z3.
(6)
Check whether the following are ring homomorphisms.
i)
ii)
iii)
f : Z  Z m : f ( z)  z m
0 0 
f : Q  M 2 (Q) : f (p)  

p p 
f : (P(X), , )  P(X) : f (A)  X \ A, 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)
7a)
b)
Prove that a prime ideal in a finite commutative ring with identity is maximal.
(3)
If f : R  S is a ring homomorphism, then show that char R  char S.
(2)
6