Question 1. The number of elements $x\in S_3$ satisfying $x^2=e$ is     .

	i)	$1$
	ii)	$2$
	iii)	$3$
	iv)	$4$

Question 2. The order of $\{k\in \mathbb{N}\mid k<9,\gcd <!--FUNCTION\; APPLICATION-->(k,9)=1\}$ under multiplication modulo $9$ is     .

	i)	$1$
	ii)	$6$
	iii)	$8$
	iv)	$9$

Question 3. Let $G$ be an abelian group and $T:G\to G$ be given by $T(x)=x^{-1}$. Then $T$ is a non-trivial homomorphism if and only if     .

	i)	$x^2\ne e$ for all $x\in G$
	ii)	$x^2=e$ for some $x\in G$
	iii)	$x^2\ne e$ for some $x\in G$
	iv)	$x^2=e$ for all $x\in G$

Question 4. A group of order     need not be abelian

	i)	$9$
	ii)	$25$
	iii)	$36$
	iv)	$49$

Question 5. A group of order $2022$ does not have a     -Sylow subgroup.

	i)	$2$
	ii)	$3$
	iii)	$5$
	iv)	none of these

Question 6. The highest power of $2$ dividing the binomial coefficient $\left(\genfrac{}{}{0ex}{}{160}{16}\right)$ is     .

	i)	$1$
	ii)	$5$
	iii)	$4$
	iv)	$10$

Question 7. If a group $G$ is internal direct product of its subgroups $N_1$ and $N_2$, then     .

	i)	$N_1\cap N_2=(e)$
	ii)	$N_1=N_2$
	iii)	$G\ne N_1{N}_2$
	iv)	$N_1\cup N_2=(e)$

Question 8. The number of invariants of a cyclic group of order $625$ is     .

	i)	$1$
	ii)	$4$
	iii)	$5$
	iv)	$25$