Question 1. Let $G$ be a finite group and $a\in G$. For which of the following $n$, $a^n=e$?

	i)	$o(G)+o(a)$
	ii)	$o(G)-o(a)$
	iii)	$\frac{o(G)}{o(a)}$
	iv)	$\frac{o(a)}{o(G)}$

Question 2. The orbit of $3$ in $(1,4,3,2)(3,5)$ is

	i)	$\{3,4,1,2\}$
	ii)	$\{1,2,3,4,5\}$
	iii)	$\{1,4,2\}$
	iv)	$\{5\}$

Question 3. The number of conjugate classes in $S_3$ is     .

	i)	$1$
	ii)	$2$
	iii)	$3$
	iv)	$6$

Question 4. A group of order     need not be abelian.

	i)	$7$
	ii)	$12$
	iii)	$25$
	iv)	$35$

Question 5. Let $G$ be a group such that $7\mid o(G)$. Then the number of $7$-Sylow subgroups in $G$ cannot be     .

	i)	$1$
	ii)	$8$
	iii)	$10$
	iv)	$15$

Question 6. The group     is simple.

	i)	${\mathbb{Z}}_2$
	ii)	${\mathbb{Z}}_2\times {\mathbb{Z}}_2$
	iii)	${\mathbb{Z}}_4$
	iv)	${\mathbb{Z}}_2\times {\mathbb{Z}}_3$

Question 7. The number of non-isomorphic abelian groups of order $16$ is     .

	i)	$2$
	ii)	$4$
	iii)	$5$
	iv)	$16$

Question 8. The invariants of the group ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ are

	i)	$2$
	ii)	$1,1$
	iii)	$2,2$
	iv)	$2,1,1$