# Classification of groups of order a product of a prime-square and another prime

## Statement

Suppose $p$ and $q$ are distinct prime numbers. This article classifies the groups of order $p^2q$.

### Case that $p$ does not divide $q - 1$ and $q$ does not divide $p - 1$

In this case, $p^2q$ is an abelianness-forcing number, i.e., all groups of this order are abelian. The two abelian groups are:

Group GAP ID second part Abelian? $p$-Sylow subgroup Is the $p$-Sylow subgroup normal? Is the $q$-Sylow subgroup normal?
cyclic group of order $p^2q$ 1 Yes cyclic group of prime-square order Yes Yes
direct product of cyclic group of order $p$ and cyclic group of order $pq$, also same as direct product of elementary abelian group of order $p^2$ and cyclic group of order $q$ 2 Yes elementary abelian group of prime-square order Yes Yes

### Case that $p$ divides $q - 1$ but $p^2$ does not divide $q - 1$ and $q$ does not divide $p^2 - 1$

In this case, there are four isomorphism classes of groups of order $p^2q$, given as follows:

Group GAP ID second part Abelian? $p$-Sylow subgroup Is the $p$-Sylow subgroup normal? Is the $q$-Sylow subgroup normal?
cyclic group of order $p^2q$ 1 Yes cyclic group of prime-square order Yes Yes
semidirect product of cyclic group of order $q$ by cyclic group of order $p^2$, where the action by conjugation of a generator is an automorphism of order $p$ 2 No cyclic group of prime-square order No Yes
semidirect product of cyclic group of order $pq$ by cyclic group of order $p$, where the action by conjugation of a generator is an automorphism of order $p$ 3 No elementary abelian group of prime-square order No Yes
direct product of cyclic group of order $p$ and cyclic group of order $pq$, also same as direct product of elementary abelian group of order $p^2$ and cyclic group of order $q$ 4 Yes elementary abelian group of prime-square order Yes Yes

### Case that $p^2$ divides $q - 1$

In this case, there are five isomorphism classes of groups of order $p^2q$, given as follows:

Group GAP ID second part Abelian? $p$-Sylow subgroup Is the $p$-Sylow subgroup normal? Is the $q$-Sylow subgroup normal?
cyclic group of order $p^2q$ 1 Yes cyclic group of prime-square order Yes Yes
semidirect product of cyclic group of order $q$ by cyclic group of order $p^2$, where the action by conjugation of a generator is an automorphism of order $p$ 2 No cyclic group of prime-square order No Yes
semidirect product of cyclic group of order $q$ by cyclic group of order $p^2$, where the action by conjugation of a generator is an automorphism of order $p^2$ 3 No cyclic group of prime-square order No Yes
semidirect product of cyclic group of order $pq$ by cyclic group of order $p$, where the action by conjugation of a generator is an automorphism of order $p$ 4 No elementary abelian group of prime-square order No Yes
direct product of cyclic group of order $p$ and cyclic group of order $pq$, also same as direct product of elementary abelian group of order $p^2$ and cyclic group of order $q$ 5 Yes elementary abelian group of prime-square order Yes Yes