Let $G$ be a group of order $21$. Find the number of non-isomorphic groups of order $21$
My solution: If the group $G$ is commutative,then $G$ can be expressed as a direct product of cyclic groups of prime power order,i.e it is either $\mathbb Z_3\times \mathbb Z_7$ or $\mathbb Z_{21}$ which is same
My problem is when $G$ is commutative.How to proceed here?
Let $p>q$ be primes. There exists a non-abelian group of order $pq$ if and only if $p\equiv 1 \bmod q$. Furthermore any tow non-abelian groups of order $pq$ are isomorphic. For a proof use the Sylow theorems. Also, one can find this result in many books. For $p=7$ and $q=3$ we obtain that there is exactly one non-abelian group of order $21$. The abelian groups are $C_{21}\simeq C_3\times C_7$.