2000

1) Let \(n\) be a fixed positive integer and let \(Z_{n}\) be the ring of integers modulo \(n\). Let \(\mathrm{G}=\left\{\bar{a} \in Z_{n} \mid \bar{a} \neq \overline{0}\right.\) and a is relatively prime to \(n\}\) . Show that \(\mathrm{G}\) is a group under multiplication defined in \(\mathrm{Z}_{\mathrm{n}}\). Hence or otherwise, show that \(a^{\phi(n)} \equiv a\) (mod \(n\)) for all integers a relatively prime to \(n\) where \(\phi\) (n) denotes the number of positive integers that are less than \(\mathrm{n}\) and are relatively prime to \(n\).

[12M]

2) Let \(M\) be a subgroup and \(N\) a normal subgroup of a group \(G\). Show that \(MN\) is a subgroup of \(G\) and MN/N is isomorphic to M/(M \(\cap\)N).

[12M]

3) Let \(F\) be a finite field. Show that the characteristic of must be a prime integer \(p\) and the number of elements in \(\mathrm{F}\) must be \(p^{m}\) for some positive integer \(m\).

[20M]

4) Let \(F\) be a field and \(F[x]\) denote the set of all polynomials defined over \(\mathrm{F}\). If \(\mathrm{f}(\mathrm{x})\) is an irreducible polynomial in \(\mathrm{F}(\mathrm{x})\), show that the ideal generated by \(\mathrm{f}(\mathrm{x})\) in \(\mathrm{F}(\mathrm{x})\) is maximal and \(\mathrm{F}[\mathrm{x}] /(\mathrm{f}(\mathrm{x}))\) is a field.

[20M]

4) Show that any finite commutative ring with no zero divisors must be a field.

[20M]

1999

1) If \(\phi\) is a homomorphism of \(G\) into \(G\) with kernel K,then show that \(K\), is a normal subgroup of \(G\).

[20M]

2) If \(p\) is a prime number and \(p^{\alpha} \vert o(G)\),then prove that G has a subgroup of of order \(p^{\alpha}\)

[20M]

3) Let \(R\) be a commutative ring with unit element where only ideal are (o) and R itself.Show that R is a field.

[20M]

4) Let \(A\) be a subset of the metric space \((M,\rho).\).If \((A,\rho)\) is compact, then show that A is closed subset of \((M,\rho)\).

[20M]

5) A sequence \(\{S_R\}\) is defined by the recursion formula \(S_{n+1}=\sqrt{3S_n,S_1}=1\).Does this sequence converge ?If so,find \(lim\;S_n\).

[20M]

TBC

6) Test for convergence the integral \(\int^1_0 x^p(log\dfrac{1}{x})^p\,dx.\)

[20M]

1998

1) Prove that if a group has only four elements then it must be abelian.

2) If \(H\) and \(K\) are subgroups of a group \(G\), then show that \(HK\) is subgroup of \(G\) if and only if \(HK=KH\).

3) Show that every group of order 15 has a normal subgroup of order 5.

4) Let \((R,+,.)\) be a system satisfying all the axioms for a ring with unity with the possible exception of \(a+b=b+a\). Prove that \((R,+,.)\) is a ring.

5) If \(p\) is prime then prove that \(Z_p\) is a field. Discuss the case when \(p\) is not a prime number.

6) Let \(D\) be a principal ideal domain. Show that every element that is neither zero nor a unit in D is a product of irreducibles.

7) Let \(X\) ba a metric space and E\(\subset\)X.Show that interior of \(E\) is the largest open set contained in \(E\), boundary of E=(closure of E)\(\cap\)(closure of X-E).

1997

1) Show that a necessary and sufficient condition for a subset Hof a group \(\mathrm{G}\) to be a subgroup is \(\mathrm{HH}^{-1}=\mathrm{H}\)

[10M]

2) Show that the order of each subgroup of a finite group is a divisor of the order of the group.

[10M]

3) In a group \(\mathrm{G}\), the commutator \((\mathrm{a}, \mathrm{b}), \mathrm{a}, \mathrm{b} \in \mathrm{G}\) is the element \(\mathrm{aba}^{-1} \mathrm{b}^{-1}\) and the smallest subgroup containing all commutators is called the commutator subgroup of G. Show that a quotient group G/H is abelian if and only if \(\mathrm{H}\) contains the commutator subgroup of \(\mathrm{G}\).

[10M]

4) If \(x^{2}=x\) for all \(x\) in a ring \(R,\) show that \(R\) is commutative. Give an example to show that the converse is not true.

[10M]

5) Show that an ideal \(\mathrm{S}\) of the ring of integers \(\mathrm{Z}\) is a maximal ideal if and only if \(\mathrm{S}\) is generated by a prime integer.

[10M]

6) Show that in an integral domain every prime element is irreducible. Give an example to show that the converse is not true.

[10M]

1996

1) Let \(R\) be the set of real numbers and \(\mathrm{G}=\{(\mathrm{a}, \mathrm{b}) \mid \mathrm{a}, \mathrm{b} \in \mathrm{R}, \mathrm{a} \neq 0\}\) \(^*\mathrm{G} \times \mathrm{G} \rightarrow \mathrm{G}\) is defined by \((a, b) *(c, d)=(a c, b e+d)\).
Show that \(\left(\mathrm{G},{ }^{*}\right)\) is a group. Is it abelian?
Is \(\left(\mathrm{H},{ }^{*}\right)\) a subgroup of \(\left(\mathrm{G},{ }^{*}\right),\) when \(\mathrm{H}=\{(1, \mathrm{b}) \mid \mathrm{b} \in \mathrm{R}\}?\)

[12M]

2) Let \(f\) be a homomorphism of a group G onto a group G’ with kernel H. For each subgroup \(\mathrm{K} '\) of G’ define \(\mathrm{K}\) by \(\mathrm{K}=\left\{\mathrm{x} \in \mathrm{G} \mid \mathrm{f}(\mathrm{x}) \in \mathrm{K}^{\prime}\right\}\) Prove that (i) \(\mathrm{K'}\) is isomorphic to \(\mathrm{K} / \mathrm{H}\). (ii) \(\mathrm{G} / \mathrm{K}\) is isomorphic to \(\mathrm{G'} / \mathrm{K'}\)

[12M]

3) Prove that a normal subgroup \(\mathrm{H}\) of a group \(\mathrm{G}\) is maximal, if and only if the quotient group \(\mathrm{G} / \mathrm{H}\) is simple.

[12M]

4) In a ring \(R\), prove that cancellation laws hold, if and only if \(R\) has no zero divisors.

[15M]

5) If \(\mathrm{S}\) is an ideal of a ring \(\mathrm{R}\) and \(\mathrm{T}\) any subring or \(\mathrm{R},\) then prove that \(\mathrm{S}\) is an ideal of \(\mathrm{S}+\mathrm{T}=\{\mathrm{s}+\mathrm{t} \mid \mathrm{s} \in \mathrm{S}, \mathrm{t} \in \mathrm{T}\}\)

[15M]

6) Prove that the polynominl \(x^{2}+x+4\) is irreducible over the field of integers modulo 11 .

[15M]

1995

1) Let \(G\) be a finite set closed under an associative binary operation such that \(\begin{array}{l} ab = ac \text { implies } b = c \\ \text { and } ba = ca \text { implies } b = c \end{array}\) for all \(a , b , c \in G .\) Prove that \(G\) is a group.

[12M]

2) Let \(G\) be a group of order \(p ^{ n },\) where \(p\) is a prime number and \(n >0 .\) Let \(H\) be a proper subgroup of \(G\) and \(N ( H )=\left\{ x \in G : x ^{-1} h x \in H \forall h \in H \right\} .\) Prove that \(N ( H ) \neq H\).

[12M]

3) Show that a group of order 112 is not simple.

[12M]

4) Let \(R\) be a ring with identity. Suppose there is an element a of \(R\) which has more than one right inyerse Prove that a has infinitely many right inverses

[15M]

5) Let \(F\) be a field and let \(p ( x )\) be an irreducible polynomial over \(F\). Let \(< p ( x )>\) be the ideal generated by \(p ( x )\). Prove that \(<p( x )>\) is a maximal ideal.

[15M]

6) Let \(F\) be a field of characteristic \(p \neq 0 .\) Let \(F ( x )\) be the polynomial ring. Suppose \(f(x)=a_{0}+a_{1} x+\ldots+a_{n} x^{n}\) is an element of \(F [ x ]\). Define \(f(x)=a_{1}+2 a_{2} x+\ldots+n a_{n} x^{n-1}\) If \(f^{\prime}(x)=0,\) prove that there exists \(g(x) \in F[x]\) such that \(f(x)=g\left(x^{p}\right)\).

[15M]