Download MTE-6-AST-2004

MTE-06
Assignment Booklet
Bachelor’s Degree Programme
Abstract Algebra
School of Sciences
Indira Gandhi National Open University
New Delhi
2004
Dear Student,
We hope you are familiar with the system of evaluation tobe followed for the Bachelor's Degree Programme. At this stage you
may probably like to re-read the section of 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.
You will also find a sample of the exam paper at the end of this booklet for your information.
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 assignments are to be submitted to your study centre coordinator within the year 2003.
Please keep a copy of your answer sheets.
Wish you good luck.
ASSIGNMENT 1
(To be done after studying Block 1 and Block 2.)
Course Code : MTE-06
Assignment Code : MTE-06/AST-1/2004
Maximum Marks : 100
1.
Which of the following statements are true? Give reasons for your answers.
viii)
The operation  : N  N  N : x  y = gcd(x, y) is associative.
For any three subsets A, B, C of a set U, A    C if and only if A  Bc  C.
The set of all mappings from {1, 2, , n} to itself form a group with respect to
composition of maps.
For any two elements a, b of a group G, o(ab) = o(ba).
The set of elements of GL2 (R) whose orders divide a fixed number n form a subgroup of
GL2 (R).
The relation ~ on subgroups of a group G, defined by ‘H~K iff H∆K’ is transitive.
If G1 is an abelian group and we can define a group homomorphism
f : G 1G2,
then G2 will also be abelian.
If 1 and 2 are in Sn such that sign (1) = sign (2), then o(1) = o(2).
ix)
x)
If G is a group, H∆G, K∆G, then G ≃ HK.
A group of order 168 has either 1 or 8 elements of order 7.
i)
ii)
iii)
iv)
v)
vi)
vii)
2. (a) For any n  N, prove that n3 + (n+1)3 + (n+2)3 is divisible by 9.
(20)
(3)
(b) Give 2 distinct elements of each of the following sets :
Zc R , N  C R, (Q N)  (N Q)
(6)
(c) Give an example each, with justification, of :
i)
a 1–1 function which is not surjective;
ii)
two functions g and f such that g o f is defined but not f o g;
iii)
a relation on Sn which is not transitive.
(6)
3. (a) For a set M, is (M, ) a group, where x  y = x  x, y  M? Is it a semigroup? Give reasons for
your answer.
(3)
(b) Give an example of a set S and two binary operations 1 and 2 on S such that (S, 1)
is a group but (S, 2) is not.
(4)
(c) Prove that for x, y, z in a group G,
[xy, z] = x[y, z] x1 [x, z] and
[z, xy] = [z, x] x [z, y] x1,
where [x, y] = xy x1 y1 is the commutator of x, y  G.
(3)
4. (a) Given a group G, let us define the set of all its elements of finite order to be the periodic part of
G.
i)
ii)
iii)
Prove that the periodic part of an Abelian group is a subgroup.
Is (i) above valid for a non-Abelian group?
Find the periodic part of the groups (C*, .) and D8.
(b) Prove that any finite subgroup of (C*, .) is cyclic.
(7)
(3)
5. (a) Find all the cosets of (C, +) with respect to the subgroup Z [ i ] = {a + ib a, b  Z}.
(1)
(b) Let gGLn(C) such that det(g)  0 and let H = SLn(C). Prove that
gH = {xGLn(C)det (x) = det g}.
(2)
(c) Partition (C, +) into disjoint right cosets of R.
(2)
6. (a) Let A and B be normal subgroups of a group G such that A  B = {e}. Prove that
 A, y  B.
(3)
xy = yx  x
(b) Prove that
i)
any subgroup of G containing [G, G] is normal in G;
ii)
if HG such that G/H is abelian, then H  [G, G].
(4)
(c) Check whether x and y are normal subgroups of
D2n = {x, yx2 = e, yn = e, xy = y1x}.
Also give two distinct elements of the corresponding quotient groups, wherever defined.
(3)
7. (a) Show that (Q, +) is not isomorphic to (Q*, .).
(4)
(b) Let S1 = {ei  R} be the circle group. Prove that R Z  S1, using the Fundamental Theorem
of Homomorphism.
Further, find the subgroup of S1 which is isomorphic to Q Z under this isomorphism.
(6)
8. (a) Write (1 2 7) (3 5)  S9 in 2-line notation. Also find its signature.
(2)
(b) Show that the product of any two transpositions in Sn can be expressed as a product of 3-cycles.
Hence prove that An is generated by all the 3-cycles in Sn.
(4)
(c) Let G be a group, H  G, X = {gHgG}.
i)
For x  G define fx : X  X : fx (gH) = xgH. Show that fx  SX.
ii)
Define f : G  SX : f (x) = fx. Show that f is a group homomorphism and
Ker f  H.
iii)
Using (ii) above, show that an infinite simple group cannot have any subgroup of finite
index.
(9)
9 (a) Show that a group of order 2002 is not simple.
(2)
(b) Give an example of a group G with normal subgroups H and K such that G/H ≃ K, G/K ≃ H but
G ≄ H  K.
(3)
ASSIGNMENT 2
(To be done after studying Block 3 and Block 4.)
Course Code : MTE-06
Assignment Code : MTE-06/AST-2/2003
Maximum Marks : 100
1.
Which of the following statements are true, and which are false? Give reasons for your answers.
i)
ii)
iii)
iv)
v)
vi)
vii)
viii)
ix)
x)
2.
The characteristic of a ring is either 0 or a prime number.
If (G, 1) is an abelian group and 2 is another binary operation defined on G, then (G, 1,
2) is a ring.
Every ring has a maximal ideal.
Every ring homomorphism is a group homomorphism.
R[x] is a PID, where R is an integral domain.
The quotient field of a ring R is a quotient ring of R which is a field.
Z11 has a unique proper subfield.
If R is a commutative ring, then x2 = xxR.
Every prime ideal of Z is maximal.
If f and g are two ring isomorphisms from R onto S, then f = g.
(20)
Let R be a ring and RR be the set of all mappings from R to itself. Show that
i)
RR is a ring with respect to pointwise addition and multiplication.
ii)
RR is commutative iff R is commutative.
iii)
RR has zero divisors.
(10)
3. (a) Give an example of a ring R and its ideal I such that R is not commutative but R
commutative.
(2)
(b) Find the centre C of the ring M3(Z), and give two distinct elements of M3(Z) C. (5)
(c) Find all the ideals of (R, +, .), where x.y = 0 x,yR.
(3)
4. (a) Which of the following are ring homomorphisms?
0 0 
i)
f : Q  M2 (Q) : f (a) = 
.
a a 
ii)
f : Z  Zn : f(z) = zn .
iii)
f : (P(X), , )  P(X) : f(A) = X \ A, where P(X) is the power set of X. (6)
(b) Use the Fundamental Theorem of Homomorphism to prove that Z5 Z2 ≃ Z3.
(5)
(c) Is Z3 Z2 an ideal of Z5 Z2? Why?
(2)
Z2 ≃ Z3 Z2.
(2)
(d) Show that (Z3 + Z2)
5. (a) Show that there doesn’t exist a non-zero ring homomorphism from Q to Z.
(3)
I is
(b) Let R be a ring with identity and I1, I2 be ideals of R such that I1 + I2 = R.
Then prove that
i)
ii)
iii)
given a1, a2R  aR such that a  ai (mod Ii) for i = 1, 2.
R R
R
 .
≃
I1  I 2
I1 I 2
The result is true more generally, viz., if I1, , In are pairwise coprime ideals of R, then
given a1, , an R  aR such that a  ai (mod Ii) i = 1, , n. Further,
R
 Ii
i 1,, n
≃
n
 R Ii . This is known as the Chinese remainder theorem (CRT).
i 1
Give an example of the use of CRT for 3 or more ideals of Z.
6. (a) Find the quotient field of R[x], where R is an integral domain.
(12)
(5)
(b) Prove that x is a prime ideal of R[x] iff R is an integral domain.
(2)
(c) Prove that a prime ideal in a finite commutative ring with identity is maximal.
(3)
7. (a) Show that Q [x] x3 – 3x2 + 15x – 6 is a field. What is its prime subfield? What is its
characteristic?
(4)
(b) Show that Z [ 2 ] is a Euclidean domain.
(6)
(c) Show that Z [  7 ] is not a PID.
(5)
8. (a) If f : R  S is a ring homomorphism, then find a relationship between char R and char S.
(2)
(b) If D is a division ring, and R a ring, then show that a non-zero homomorphism
injective.
(3)
f : D  R is
SAMPLE PAPER
MTE-6 : ABSTRACT ALGEBRA
Time : 2 hours
Maximum Marks : 50
Note : Q. 1 is compulsory. Attempt any four from remaining questions (Q. 2 to 7).
1.
2.
3.
4.
5.
Which of the following statements are true? Justify your answers.
10
(a)
(b)
(c)
(d)
(e)
If a and b are elements of a group G such that O(a) = 2 and O(b) = 3, then O(ab) = 6.
If (a, b)  A  B, then a  A and b  B.
In a ring with unity, the sum of any two units is a unit.
If R is an integral domain and I is an ideal of R, char (R) = char (R/I).
In a commutative ring every prime ideal is a maximal ideal.
(a)
Use Sylow theorems to determine all the groups of order 6.
(b)
Check whether the ring
(a)
(i)
Prove that
V4 = {I, (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)} is a normal subgroup of A4.
(ii)
Let K = {I, (1 2 3), (1 3 2)}. Check whether V4K is a subgroup of A4. Also find
the number of elements in V4K.
6
5
F[X]
is a field or not, where F = Z11. Also find the
 X X  4 
number of elements in this ring.
5
2
(b)
Show that 2  5 is an irreducible element of the ring Z[ 5] {a  b 5 a, bZ} but
it is not a prime element.
4
(a)
Let R be a UFD and A1  A2    An   be a chain of ideals in R. Check
whether there is an integer k such that Ak = Ak+1 = 
3
(b)
Prove by induction that 2n  2n + 3 for each n  4.
(c)
Let R and S be commutative rings and f : R  S be a ring homomorphism. Show that the
inverse image of a prime ideal in S under f is a prime ideal of R.
4
  x x 

x C, x  0  . Check whether G is an abelian group under
Let G   

 0 0 

matrix multiplication or not.
5
(a)
3
(b)
Let X be an infinite set and I the set of finite subsets of X. Show that I is an
ideal of P(X), the ring of all subsets of X.
3
(c)
Express the following as a product of disjoint cycles :
1 2 3 4 5 6 7 8 

.
8 4 7 2 1 3 6 5
6.
7.
2
(a)
If H and K are normal subgroups of a group G such that H  K = {e}, show
that hk = kh for all h  H, k  K.
2
(b)
Show that for m, n  Z, nZ  mZ if and only if mn. Hence obtain all the
ideals of Z20.
4
(c)
Let U be an ideal of a ring R.
Define [R : U] = {x  R rx  U}. Prove that
(i)
[R : U] is an ideal of R.
(ii)
U  [R : U]
4
(a)
Consider the relation  defined on Z by n  m if and only if mn  0. Check
whether this is an equivalence relation.
3
(b)
Show that
(d)
Find the order of the group G of inner automorphisms of Q8, the group of
quaternions. Is G abelian? Why?
Q[x]
is a field.
 8x  6x 2  9x  24 
3
3
4