2008

1) If a group is such that \((a b)^{2}=a^{2} b^{2}\) for all \(a, b \in G\), then prove or disprove that \(\mathrm{G}\) is abelian.

[10M]

2) Prove or disprove that there exists an integral domain with six elements.

[10M]

3) Prove or disprove that \(\left(R^{*}, .\right)\) is isomorphic to \((R,+)\).

[13M]

4) Find the sylow subgroups of the group \(Z_{24}\) (the additive group of modulo 24).

[10M]

2007

1) Prove or disprove that if \(H\) is a normal subgroup of a group \(G\) such that \(H\) and \(G / H\) are cyclic, then \(G\) is cyclic.

[5M]

2) Show by counter-example that the distributive laws in the definition of a ring is not redundant.

[5M]

3) In the ring of integers modulo 10 ( \(i \cdot e_{\text {। }} Z_{10} \oplus_{10} \cdot \Theta_{10}\) ) find the subfields.

[5M]

4) Prove or disprove that only non-singular matnices form a group under matrix multiplication.

[5M]

5) Show that there are no simple groups of order 63 and 56 .

[10M]

6) Prove that every Euclidean domain is PID.

[10M]

2006

1) Show that the set of cube roots of unity is a finite Abelian group with respect to multiplication z.

[10M]

2) Prove that the set of all real numbers of the form \((a+b \sqrt{b}),\) where a and \(b\) are rational numbers, is a field under usual addition and multiplication.

[10M]

3) Show that the set \(S=\{1,2,3,4 \}\) forms an Abelian group under the operation of multiplication modulo 5 as defined below: |\(x\;mod\;5\)| 1 | 2 | 3 | 4 | |———–|—-|—-|—-|—-| |1 | 1 | 2 | 3 | 4 | |2 | | | | 3 | |3 | | | 4 | 2 | |4 | | 3 |4h 2 | 1 |

[10M]

2005

1) Show that the set of cube roots of unity is a finite Abelian group with respect to multiplication.

[10M]

2) Show that the set \(S =\{1,2,3,4\}\) forms an Abelian group for the operation of maltiplication modulo 5.

[10M]

3) Prove that the set of all real numbers of the form \(a+b \sqrt{2},\) where \(a\) and \(h\) are real numbers, is a field under the usual addition and multiplication.

[10M]

4) If \(R\) is commutative ring with unit element and \(M\) is an ideal in \(R\), then show that \(M\) is maximal ideal if R/M is a field

[10M]