IAS PYQs 2
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 \(f(x)=\dfrac{f_{0}(x)}{a}\) 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 \(\left( x ^{2}+1\right)\) in \(Z _{3}[ x ]\)
[10M]
6) Let \(Q\) be the set of rational number and \(Q\left(2^{1 / 2}, 2^{13}\right)\) the smallest extension field of \(Q\) containing \(2^{1 / 2}, 2^{1 / 3}\). Find the basis for \(Q\left(2^{1 / 2}, 2^{13}\right)\) 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 \(\dfrac{F[x]}{<x^{2}+1>}\) and \(\dfrac{F[x]}{x^{2}+x+4}\) are isomorphic fields each having 121 elements.
[10M]
6) Find the degree of splitting field \(x^{5}-3 x^{3}+x^{2}-3\) 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 \(\dfrac{G}{C}\) 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 \(x y x^{-1} y^{-1}\) 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 \(\dfrac{A}{I}\) is commutative.
6) If a finite field of characteristic \(\mathrm{p}\) has \(q\) elements show that \(q=p^{n}\) for some \(\mathrm{n}\).