2008

1) If a group is such that $(ab{)}^2=a^2{b}^2$ 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 $(R^{\ast },.)$ 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 ( $i\cdot e_{\text{। }}{Z}_{10}{\oplus }_{10}\cdot {\mathrm{\Theta }}_{10}$ ) 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: |$xmod5$| 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]