1988

1) If H and K are normal subgroups of a group G such that H∩K={e}, show that hk=kh for all h∈H and k∈K.

2) Show that the set of even permutations on n symbols, n>1, is a normal subgroup of the symmetric group Sn and has order n!2.

3) Show that the numbers 0,2,4,6,8 with addition and multiplication modulo 10 form a field isomorphic to Js, the ring of integers modulo 5. Give the isomorphism explicity.

4) R is a commutative ring with identity and U is an ideal of R. show that the quotient ring R/U is a field if and only if U is maximal.

1987

1) Let f:X→Y and g: Y→Z be both bijections, prove that gof is bijection and (g∘f)−1=f−1∘g−1.

2) Prove that HoK is a subgroup of (G,o) if and only if H∘K=K o H.

3) If G is a finite group of order g and H is a subgroup of G of order h, then prove that h is a factor of g.

1986

1) Prove that a map f:X→Y is injective iff f can be left cancelled in the sense that f∘g=f∘h⇒g=h. f is subjective iff it can be right cancelled in the sense that g∘f=h∘f⇒g=h.

2) The product HK of two sub groups H,K of a group G is a sub group of G if and only if HK=KH.

3) Prove that a finite integral domain is a field.

1985

1) State and prove the fundamental theorem of homomorphism for groups.

2) Prove that the order of each subgroup of a finite group divides the order of the group.

3) Write if each of the following statements is true or false:
(i) If a is an element of a ring (R,+,.) and m and n∈N, then (am)n=amn.
(ii) Every sub group of an abelian group is not necessarily abelian.
(iii) A semi group (G,.) in which the equations ax=b and xa=b are solvable (for any a,b) is a group.
(iv) The relation of isomorphism in the set of all groups is not an equivalence relation.
(v) There are only two abstract groups of order six.

1984

1) Prove that a non-void subset S of a ring R is a sub ring of R, if and only if, a−b∈S and ab∈S for all a,b∈S.

2) Prove that an integral domain can be embedded in a field.

3) Prove that for any two ideals A and B of a ring R, the product AB is an ideal of R.

1983

1) Show that the set I×I of integers is commutative ring with respect to addition and multiplication defined as follows. (a,b)+(c,d)=(a+c,b+d)
(a,b)⋅(c,d)=(ac,bd)
where a, b, c, d∈I

2) Prove that the relation of isomorphism in the set of all groups is an equivalence relation.

3) Prove that a polynomial domain K[x] over a field K is a principal ideal domain.