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 ${\displaystyle \frac{n!}{2}}$.

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 $f:X\to Y$ and $g$: $Y\to Z$ 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 $f:X\to Y$ 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 $\mathrm{H}\mathrm{K}=\mathrm{K}\mathrm{H}$.

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 ${\left(a^m\right)}^n=a^{mn}$.
(ii) Every sub group of an abelian group is not necessarily abelian.
(iii) A semi group $(\mathrm{G},.)$ in which the equations $\mathrm{a}\mathrm{x}=\mathrm{b}$ 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 $\mathrm{A}\mathrm{B}$ 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. $(a,b)+(c,d)=(a+c,b+d)$
$(a,b)\cdot (c,d)=(ac,bd)$
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.