Abstract Algebra Answers

1.a) Obtain 3 distinct cosets of H = { I, (1 2) (3 4), (1 4) (2 3}, (1 3) (2 4)} in S4. Justify your answer.

b) Prove that if R is a finite commutative ring with identities, then every prime ideal of R is a maximal ideal
.
c) Check if the following are ring homomorphisms :

i) f: M2 (Z) -> Z : f ( a b ) = a .
( c d )
ii) f: {( a b ) I a b d belongs to Z} -> Z*Z :
0 d

f:( a b ) = ( a, d )
0 d

1. a) Let G be a group of order 36 with 4 Sylow- 3 subgroups. Show that the intersection of any two of
these subgroups is of order 3.

(b) Prove by induction that 2n > 2n + 3 v n > 4.

(c) Express the following permutation as the product of disjoint cycles:
( 1 2 3 4 5 ) ( 2 4 7 )
( 3 1 5 2 4) (4 7 2 )

1. a) Let C and H be groups of orders p, q where p and q are distinct primes. Show that any homomorphism from G to H must take any element of G to the identity element of H.

b) Show that if R.is a commutative ring with no zero divisors, then the polynomial ring R[X] has no zero divisors.

c) Find the quotient field of z[ root under 3 ] = {a + b(root under 3) la, b belongs to z}.

.

a) Let Z2 be the field with 2 elements .Let G denote the set of all 3 x 3 matrices of the form
( 1 0 a )
( 0 1 b )
(0 0 1)
with a, b e 22. Show that G is a group of order 4 with respect to matrix multiplication .

( b ) (i) let f :R -> S be an onto ring homomorphism. Show that if I is an ideal of R, then f(I) is an ideal of

(il) If f were not onto, would f(l) still be an ideal of S ? Give reasons for your answer,

(a) Count the number of distinct 3 - cycles in S4. Further, find the number of distinct 4 cycles of length r is Sn, where r>1.
(b) If R is a PID, so is R[x]. True or false ? why ?

(c) Check whether or not '—' is an equivalence relation on Z, where 'a--b if a = br for some n€N'.

Which of the following statements are true ? Give 10
reasons for your answer.

(a) Subring of a UFD must be a UFD.

(b) If K is a normal subgroup of H and H is a normal subgroup of G, then K is a normal subgroup of G.

(c) The field of quotients of Z + Z is R.

(d) {Z, Q, IGNOU} is a set.

(e) Any subset of a ring (R, ®) is a ring with respect to the operations of 0 and C!).

(a) Let G be a group of order 21. Show that G has a proper normal non-trivial subgroup.

(b) Apply the principle of induction to show that n3 + (n + 1)3 + (n + 2)3 is divisible by 9 V nEN.

(c) State the Fundamental theorem of Algebra. Also give a polynomial of degree n over a ring R which has more than n roots in R.

(a) Show that the group G = {I, 3, 5, under multiplication modulo 8 is isomorphic to the group H = ri, 5, 7, T11 u nder multiplication modulo 12. Also show that neither of them is isomorphic to the group F= {I, 3, 7, 9} under multiplication modulo 10.

(b) Check whether Q [x](4x7 — 3x5+ 3x4 — 15) is a field or not. If it is a field, give its characteristic. If
it is not a field, obtain its quotient field.

(c) Let R and R' be commutative rings and f : R--->Rr be a ring homomorphism. If I is an
ideal of R, check whether f(I) is an deal of
R' or not.

(a) If H and K are subgroups of a group G, and 4 if only H is a normal subgroup of G, then prove or disprove that HK is a subgroup of G. Give an example to show that HK need not be a subgroup if neither H nor K is a normal subgroup of G.

(b) Show that (I) : R[x]-->R, defined by ((ao + + + a nxn) ao + + + an, is a ring homomorphism. Check whether ker (I) is a principal ideal or not. Is it a maximal ideal ? Why, or why not ?

(a) Prove that any finite group is a subgroup of 6 a permutation group.

(b) Let d be a Euclidean norm on a Euclidean domain D. Show that if sEZ such that s + d(1)>O, then g : D\ (0}—>Z g(a) = d(a) + s, for non-zero aED, is a Euclidean norm on D.