1994

1) If G is a group such that (ab)n=anbn for three consecutive integers n for all a, b in G, then prove that G is abelian.

[10M]

2) Can a group of order 42 be simple? Justify your claim.

[10M]

3) Show that the additive group of integers modulo 4 is isomorphic to the multiplicative group of the non-zero elements of integers modulo 5. State the two isomorphisms.

[10M]

4) Find all the units of the integral domain of Gaussian integers.

[10M]

5) Prove or disprove the statement: The polynomial ring I [x] over the ring of integers is a principal ideal ring.

[10M]

6) If R is an integral domain (not necessarily a unique factorization domain) and F is its field of quotients, then show that any element f(x) in F(x) is of the form f(x)=f0(x)a where f0(x)∈R [ x ],a \in R$$

[10M]

1993

1) If G is a cyclic group of order n and p divides n, then prove that there is a homomorphism of G onto a cyclic group of order p. what is the Kemel of homorphism?

[10M]

2) Show that a group or order 56 cannot be simple.

[10M]

3) Suppose the H,K are normal subgroups of a finite group G with H a normal subgroup of K. If P=K/H,S=G/H, then prove that the quotient groups S/P and G/K are isomorphic.

[10M]

4) If Z is the set of integers then show that z[√−3]={a+√−3b;a,b∈Z} is not a unique factorization domain.

[10M]

5) Construct the addition and multiplication table for Z3[x]/<x2+1> where Z3 is the set of integers modulo 3 and <x2+1> is the ideal generated by (x2+1) in Z3[x]

[10M]

6) Let Q be the set of rational number and Q(21/2,213) the smallest extension field of Q containing 21/2,21/3. Find the basis for Q(21/2,213) over Q.

[10M]

1992

1) If H is a cyclic normal subgroup of a group G, then show that every subgroup of H is normal in G.

[10M]

2) Show that no group of order 30 is simple.

[10M]

3) If p is the smallest prime factor of the order of a finite group G, prove that any subgroup of index p is normal.

[10M]

4) If R is a unique factorization domain, then prove that any f∈R[x] is an irreducible element of R[x], if and only if either f is an irreducible element of R or f is an irreducible polynomial in R[x}.

[10M]

5) Prove that x2+1 and x2+x+4 are irreducible over F, the field of integers modulo 11 . Prove also that F[x]<x2+1> and F[x]x2+x+4 are isomorphic fields each having 121 elements.

[10M]

6) Find the degree of splitting field x5−3x3+x2−3 over Q, the field of rational numbers.

[10M]

1991

1) If the group G has no non-trivial subgroups, show that G must be finite of prime order.

2) Show that a group of order 9 must be abelian.

3) If the integral domain D is of finite characteristic, show that the characteristic must be a prime number.

4) Find the greatest common divisor in J(i) of

(i) 3+4i and 4−3i (ii) 11+7i and 18−i

5) Show that every maximal ideal of a commutative ring R with unit element must be a prime ideal.

1990

1) Let G be a group having no proper subgroups. Show that G should be a finite group of order which is a prime number or unity.

2) If C is the centre of a group G and GC is cyclic, prove that G is abelian.

3) Show that the set of Gaussian integers is a Euclidean ring. Find an HCF of the two elements 5i and 3+i.

1989

1) Let G be a group of order 2p,p being a prime. Show that there exist a normal subgroup of G of order p.

2) Give an example of an infinite group in which every element is of finite order.

3) Let G be a group. Consider the set of elements of the form xyx−1y−1 where x and y are in G. If H is the smallest subgroup of G containing all these elements, show that H is a normal subgroup of G and that the factor group G/H is abelian.

4) If each element of a ring is idempotent, show that the ring is commutative.

5) Let A be a ring and I be a two sided ideal generated by the subset of all elements of the form a b−ba; a, b∈A, prove that the residue class ring AI is commutative.

6) If a finite field of characteristic p has q elements show that q=pn for some n.