1988

1) If \(\mathrm{H}\) and \(\mathrm{K}\) are normal subgroups of a group \(\mathrm{G}\) such that \(H \cap K=\{e\},\) show that \(h \mathrm{k}=\mathrm{k} \mathrm{h}\) for all \(h \in H\) and \(k \in K\).

2) Show that the set of even permutations on \(\mathrm{n}\) symbols, \(\mathrm{n}>1\), is a normal subgroup of the symmetric group \(S_{n}\) and has order \(\dfrac{n!}{2}\).

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

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

1987

1) Let \(f: X \rightarrow Y\) and \(g\): \(Y \rightarrow Z\) be both bijections, prove that gof is bijection and \((g \circ f)^{-1}=f^{-1} \circ g^{-1}\).

2) Prove that HoK is a subgroup of \((\mathrm{G}, \mathrm{o})\) if and only if \(H \circ K=K\) o \(H\).

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

1986

1) Prove that a map \(f: X \rightarrow Y\) is injective iff \(\mathrm{f}\) can be left cancelled in the sense that \(f \circ g=f \circ h \Rightarrow g=h\). \(f\) is subjective iff it can be right cancelled in the sense that \(g \circ f=h \circ f \Rightarrow g=h\).

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

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 \((\mathrm{R},+, .)\) and \(\mathrm{m}\) and \(n \in N,\) then \(\left(a^{m}\right)^{n}=a^{m n}\).
(ii) Every sub group of an abelian group is not necessarily abelian.
(iii) A semi group \((\mathrm{G}, .)\) in which the equations \(\mathrm{ax}=\mathrm{b}\) and \(\mathrm{x} \mathrm{a}=\mathrm{b}\) are solvable (for any \(\mathrm{a}, \mathrm{b}\)) 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 \(\mathrm{R}\) is a sub ring of \(\mathrm{R},\) if and only if, \(a-b \in S\) and \(ab \in S\) for all \(a, b \in S\).

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

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

1983

1) Show that the set \(I \times I\) of integers is commutative ring with respect to addition and multiplication defined as follows. \((a, b)+(c, d)=(a+c, b+d)\)
\((a, b) \cdot(c, d)=(a c, b d)\)
where \(a\), \(b\), \(c\), \(d \in I\)

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 \(\mathrm{K}\) is a principal ideal domain.