Sylow's theorems

Find the number of Sylow 5-subgroups in $A_{5}$.

Verify that the set $P$ of matrices

over $Z_{3}$ forms a group isomorphic to the quaternion group $Q_{8}$ and being a Sylow 2-subgroup of $\mathrm{SL}\left(2, Z_{3}\right)$. Show that $P \triangleleft \mathrm{SL}\left(2, Z_{3}\right)$.

Show that the groups $S_{4}$ and $\mathrm{SL}\left(2, Z_{3}\right)$ are not isomorphic. Are the groups $\operatorname {PSL}\left(2, Z_{3}\right)$ and $A_{4}$ isomorphic?

Prove that every group $G$ of order $p q$ ($p<q$ prime numbers) is either cyclic, or nonabelian with a normal Sylow $q$-subgroup, and the latter is possible if and only if $q-1$ is divisible by $p$. In particular, all groups of order 15 are cyclic.

Obtain anew (see [ВА I, Ch. $6 \S 1$, Example]) the congruence $(p-1) !+1 \equiv 0(\bmod p)$ for prime $p$ by direct computation of the number $N_{p}$ of Sylow $p$-subgroups in the symmetric group $S_{p}$.

Verify that a group $G$ of order $\left|G\right| \leqslant 30$ cannot be simple.