Question 1. The set of all functions from a nonempty set $S$ to itself is     group under the operation of composition of functions.

	i)	not a
	ii)	a trivial
	iii)	a non-abelian
	iv)	an abelian

Question 2. If $13^{39}\equiv x(mod11)$, then $x=$     .

	i)	$6$
	ii)	$2$
	iii)	$1$
	iv)	$10$

Question 3.     is the class equation of a group of order $5$.

	i)	$1+2+2$
	ii)	$2+3$
	iii)	$1+1+1+1+1$
	iv)	$5$

Question 4. The number of permutations in $S_5$ that commute with $(1,2)$ is     .

	i)	$10$
	ii)	$12$
	iii)	$20$
	iv)	$60$

Question 5. The order of a $2$-Sylow subgroup of the alternating group $A_6$ is     .

	i)	$3$
	ii)	$4$
	iii)	$8$
	iv)	$16$

Question 6. The highest power of $3$ dividing the binomial coefficient $\left(\genfrac{}{}{0ex}{}{486}{27}\right)$ is     .

	i)	$9$
	ii)	$2$
	iii)	$3$
	iv)	$1$

Question 7. The invariants of ${\mathbb{Z}}_4\times {\mathbb{Z}}_2\times {\mathbb{Z}}_8\times {\mathbb{Z}}_2$ are    

	i)	$16$
	ii)	$4,2,8,2$
	iii)	$2,2,2,2$
	iv)	$3,2,1,1$

Question 8. The number of non-isomorphic non-cyclic abelian groups of order $32$ is     .

	i)	$7$
	ii)	$5$
	iii)	$1$
	iv)	$6$