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
2007
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, 2007,
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-1/2007
Total Marks : 100
1. Which of the following statements are true? Give reasons for your answers.
i)
If H and K are disjoint normal subgroups of G such that G = HK, then G = H  K.
ii)
For any two sets H and K, H \ K = H  (HKc).
iii)
If G is a group with respect to addition, then it is also a group with respect to subtraction.
iv)
(R \ { 0 }, .)  (R, +).
v)
Every subgroup of an infinite group is infinite.
vi)
For a  (G, .), am + an = am + n, where m, n  N.
vii)
If G is a group, H  G, and x, y  G are such that  z  Hx such that z  Hy, then
Hx  Hy = .
viii) Every group of order pn, where p is a prime and n  N, is of the form G1  G2  …. Gn,
where Gi= p.
ix)
Every permutation in Sn is a product of disjoint transpositions, where n  2.
x)
If H and K are subgroups of G such that H  K  G, then H  G and K  G.
(20)
2. a) Find the domain and range of the following functions:
 1 , x A n
i) f, defined by f(x) = 
.
0, x Sn \ A n
ii) The inverse of the function f, defined by f(x) = 2x – 3.
Also check if the functions are 1 – 1 or not.
(4)
b) Define a relation  on a group G by 'a  b if and only if a = h b k, for some h  H, k  K,
where H  G, K  G. Check whether  is an equivalence relation or not.
(3)
c) Show that n3 + (n+1)3 + (n+2)3 is divisible by 9  n  N.
(3)
 cos t sin t 



3. a) Consider M = 
 t R  . Check whether or not M is an abelian group with


  sin t cos t 

respect to matrix multiplication. Further, find an element in M of order 3.
(6)
b) Let H  Aut G, where (G, .) is a group. Consider the set H  G together with the binary
operation  , defined by
(f1, g1)  (f2, g2) = (f1
f2, f 21 (g1) . g2).
Check whether (H  G, ) is a group or not.
3
(4)
4. a) Prove that if G is a finite group and H  G such that hh'  H whenever h, h'  H, then H  G.
(4)
b) Prove that if G is a cyclic group and do(G), then G has one and only one subgroup of order
d.
(6)
G
is cyclic, then G is abelian. Will this be true if Z(G) is
Z(G)
replaced by any normal subgroup H of G? Give reasons for your answer.
(4)
5. a) Let G be a group. Show that if
b) Prove that Aut (Zn)
U (Zn), the group of units of Zn.
(6)
6. a) Which of the following pairs of groups are isomorphic? Give reasons for your answers.
S3 and Z 6 Z .
ii) Z 6Z and Z 2Z  Z 3Z .
i)
(4)
b) Find the minimum number of elements a non-abelian group can have.
(3)
c) Let G be a group. For each g  G, define fg : G  G : fg (x) = gx.
Show that fg  Aut G  g  G.
(3)
7. Let G be a group, H1 and H2 be subgroups of G and N1  H1, N2  H2. Then prove that
i) N1 (H1  N2)  N1 (H1  H2)
ii) (H1  N2) (H2  N1)  H1  H2
iii) N1 (H1  H2) / N1 (H1  N2) (H1  H2) / (H1  N2) (H2  N1)
(15)
8. a) Consider S1 = {z  C   z  = 1}. Let R+ = { r  R  r > 0}. Show that
C*

ii) C
i)
S1  R+
R+
S1, using the Fundamental Theorem of Homomorphism.
(8)
b) Find the structure of a group of order 77.
(4)
 x   x  ut 
d) For each u  R, consider gu : R2  R2 : g u    
.
t  t 
Prove that G = {gu  u  R} is a group.
(3)
4
ASSIGNMENT 2
(To be done after studying Blocks 1 and 2.)
Course Code : MTE-06
Assignment Code : MTE-06/TMA-2/2007
Total Marks : 100
1. Which of the following statements are true? Give reasons for your answers.
i)
If R is a Euclidean domain, so is R  R.
ii)
Every abelian group can be turned into a ring.
iii)
If I is a proper ideal of a ring R with identity, then 1  I.
iv)
If R is a UFD, so is R
v)
The characteristic of a field is either 0 or .
vi)
iZ is a ring with respect to the usual addition and multiplication.
vii)
If e is the identity of a ring R, then it is also the identity of R
I
for every ideal I or R.
I
for every ideal I of R.
viii) Given any n  N, we can find a commutative ring with only n elements in it.
ix)
Every polynomial in k[x], of degree n, has n roots in k, where k is a field.
x)
If the product of two polynomials a(x) and b(x) in Z[x] is primitive, then both a(x) and b(x)
are primitive.
(20)
2. a) Check whether (Q, , ) is a ring, where
a  b = ab and a
b=a+b
b) Check whether (Z,  ,
a  b = a+b+1 and a
 a, b  Q.
(2)
) is a ring, where
b = a + b + ab
 a, b  Z.
c) Check whether (P (X), , ) is a ring, where P(X) is the power set of X.
3. a) Let (R, +, ) be a ring. Define  and
(2)
on R  Z by
(a, m)  (b, n) = (a + b, m + n) and (a, m)
i)
(6)
(b, n) = (ab + mb + na, mn).
Check whether R  {0} and R  {1} are ideals of R  Z.
ii) Show that R
R  {0}.
(7)
5
b) Consider R = ({a, b, c, d},  ,

a
b
c
d
a
a
b
c
d
b
b
a
d
c
c
c
d
a
b
), where  and
d
d
c
b
a
are defined by the tables below:
a
b
c
d
a
a
a
a
a
b
a
b
c
d
c
a
a
a
a
Check whether S1 = {a, b} and S2 = {a, b, c} are subrings of R.
d
a
b
c
d
(3)
4. a) Consider n : Z  Z : n (z) = nz. Under what conditions on n is n a
i) group homomorphism?
ii) ring homomorphism?
(4)
b) Which of the following are ring homomorphisms? For those that are, find their kernels and
images.
i)
f : M3 (R)  R : f (A) = det (A)
ii) f : Z  Z  Z  Z : f (a, b) = (3a, 5b)
 n
 n
iii) f : C [x]  C : f  a i x i   a i
 i 0
 i 0




 n

iv) f : Q [x] \ {0}  Q : f  a i x i  = an.
 i0




(7)
c) Let f : R  S be a surjective ring homomorphism. Show that the image of Z(R) under f is
contained in Z(S), where for any ring A, Z(A) = {x  A  xa = ax  a  A} denotes the
centre of A.
Would f (Z(R))  Z (S) still hold if f were not surjective? Give reasons for your answer.
(4)
5. a) Let R be a commutative ring with unity, and I and J be ideals of R such that I + J = R. Show
that for any a, b  R  x  R such that x  a  I, x  b  J.
(Hint: Since 1  R,  i  I, j  J with 1 = i + j. Use this to prove the result.)
b) Obtain all the group and ring homomorphisms from Z to Q.
6. a) Let F be a field. Prove that F[x]
F, where a  F.
 x a 
(5)
(5)
(6)
b) Show that the zero divisors in the ring of all functions on [0, 1] are precisely the functions that
vanish at some point.
(4)
6
7. a) Which of the following are field extension of Q, and why?
Q[x]
i) Q  Q
ii)
< 2x 3  6x +3 >
C[x]
iii) Q[x] 2
iv)
< x +7 
 x 1 
(4)
b) Prove that an ideal I of a ring R is prime if and only if I is a kernel of a homomorphism from R
to a field.
(4)
c) Find two elements of End A which do not commute with each other, where A=(R[x], +).
(2)
8. a) Apply Eisenstein’s criterion to check whether 3x2 + 30x – x3 + 43 is irreducible over Z.
(Hint: p(x) is irreducible over Z if and only if p (x  2) is irreducible over Z.)
b) Find the quotient field of R = Z[x]
< x2  1 
. Also write two distinct elements of R.
(3)
(7)
9. a) Let I and J be ideals of a ring R, and P a prime ideal of R containing I  J. Show that either
I  P or J  P.
(3)
b) Check whether or not  1 x,3  is a principal ideal of R[x]
7
 x 3 1 
.
(2)