2004

1) If every element except the identity, of a group is of order \(2,\) Prove that the group is abelian.

2) Prove that the set \(R=\{a+\sqrt{2} b, a, b \in I\}\) is a ring. Is it an integral domain? Justify your answer.

3) Let \(G\) be a group of real numbers under addition and \(G'\) be a group of +ve real numbers under multiplication. Show that the mapping \(f: G \rightarrow G^{\prime}\) definedby \(f(x)=2^{a} \quad \forall a \in G\) is a homomorphism. Is it an isomorphism too? Supply reasons.

2003

1) Let \(G=\{a \in R:-1<a<b\}\). Define a binary operation \(^{*}\) on \(G\) by \(a^{*} b=\dfrac{a+b}{1+a b}\) for all \(a, b \in G\). Show that \(\left(G,^{*}\right)\) is a group.

2) Let \(R\) be the set of matrices of the form \(\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right], a, b \in F,\) Where \(F\) is a field with usual addition and multiplication as binary operations. Show that \(R\) is a commutative ring with unity. Is it a field if \(F=z_{2}, z_{5} ?\)

2002

1) Show that every group consisting of four or less than four elements is abelian.

2) In the symmetric group \(S_{n}\) of Permutations of \(n\) symbols, find the number of even permutation. Show that the set \(A_{n}\) of even permutations forms a finite group. Identify \(S_{n}\) and \(A_{n}\) when \(n=4.14\)

3) If \(F\) is a finite field & \(\alpha, \beta\) are two non-zero elements of \(F\), then show that there exist elements \(a\) & \(b\) in \(F\) such that \(1+\alpha a^{2}+\beta b^{2}=0\).

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

2001

1) Write the elements of the symmetric group \(S_{3}\) of degree 3 , Prepare its multiplication table and find all normal subgroups of \(S_{3}\).

2) If every element of a group \(G\) is its own inverse, Prove that the group \(G\) is abelian. Is the converse true ? Justify your claim.

3) Define a unique factorization domain. Show that \(z[\sqrt{-5}]\) is an integral domain which is not a unique factorization domain.

2000

1) Show that an infinite cyclic group is isomorphic to the additive group of integers.

2) Show that every finite integral domain is a field.

3) Show that every finite field is a field extension of field of residues modulo a prime \(P\).