Commutator subgroup of $\mathbb{Z}_{p^2} \rtimes_{\varphi} \mathbb{Z}_q$ with $p,q$ distinct primes

Question

Can someone please help to find the commutator subgroup and its order for the group $\mathbb{Z}_{p^2} \rtimes_{\varphi} \mathbb{Z}_q$, where $p, q$ are distinct primes?

Answer 1

I suggest you do not work with the definition of the commutator subgroup directly, but rather the equivalent definition: it is the smallest normal subgroup $N$ (w.r.t. inclusion) such that the factor group $G/N$ is commutative.

The $p$-Sylow $\mathbb{Z}_{p^2}$ is a normal subgroup such that the factor group is isomorphic to $\mathbb{Z}_{q}$, hence is commutative. So the commutatator subgroup is contained in $\mathbb{Z}_{p^2}$. The only thing you need to check is if there is a normal subgroup inside $\mathbb{Z}_{p^2}$. It would be closed under conjugation by an element of order $q$. If there is such a thing, you still have to check the factor group. It is easy to figure out the isomorphism type of it.