1994

1) If $G$ is a group such that $(ab{)}^n=a^n{b}^n$ 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 $f(x)={\displaystyle \frac{f_0(x)}{a}}$ where $f_0(x)\in$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 $z[\sqrt{-3}]=\{a+\sqrt{-3}b;a,b\in Z\}$ is not a unique factorization domain.

[10M]

5) Construct the addition and multiplication table for $Z_3[x]/<x^2+1>$ where $Z_3$ is the set of integers modulo 3 and $<x^2+1>$ is the ideal generated by $(x^2+1)$ in $Z_3[x]$

[10M]

6) Let $Q$ be the set of rational number and $Q(2^{1/2},2^{13})$ the smallest extension field of $Q$ containing $2^{1/2},2^{1/3}$. Find the basis for $Q(2^{1/2},2^{13})$ over $Q$.

[10M]

1992

1) If $\mathrm{H}$ is a cyclic normal subgroup of a group $\mathrm{G},$ then show that every subgroup of $\mathrm{H}$ is normal in $\mathrm{G}$.

[10M]

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

[10M]

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

[10M]

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

[10M]

5) Prove that $x^2+1$ and $x^2+x+4$ are irreducible over $F$, the field of integers modulo 11 . Prove also that ${\displaystyle \frac{F[x]}{<x^2+1>}}$ and ${\displaystyle \frac{F[x]}{x^2+x+4}}$ are isomorphic fields each having 121 elements.

[10M]

6) Find the degree of splitting field $x^5-3x^3+x^2-3$ 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 $\mathrm{D}$ 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 $\mathrm{R}$ 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 $\mathrm{C}$ is the centre of a group $\mathrm{G}$ and ${\displaystyle \frac{G}{C}}$ is cyclic, prove that $\mathrm{G}$ is abelian.

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

1989

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

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 $xy{x}^{-1}{y}^{-1}$ where $x$ and $y$ are in $G$. If $H$ is the smallest subgroup of G containing all these elements, show that $\mathrm{H}$ is a normal subgroup of $\mathrm{G}$ and that the factor group $\mathrm{G}/\mathrm{H}$ 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 $\mathrm{b}-\mathrm{b}a$; $\mathrm{a}$, $\mathrm{b}\in \mathrm{A}$, prove that the residue class ring ${\displaystyle \frac{A}{I}}$ is commutative.

6) If a finite field of characteristic $\mathrm{p}$ has $q$ elements show that $q=p^n$ for some $\mathrm{n}$.