Converse of congruence condition for prime power Sylow numbers

Statement

Suppose $p,q$ are distinct primes and $r$ is a natural number such that:

$q^{r}\equiv 1\mod p$.

Then, there exists a finite group in which the number of $p$-Sylow subgroups (i.e., the $p$-Sylow number) equals $q^{r}$.

Related facts

• Converse of congruence condition for Sylow numbers for the prime two
• Congruence condition on Sylow numbers

Proof

Proof idea

The idea is to take the semidirect product of the additive group of the field with $q^{r}$ elements by a cyclic subgroup of order $p$ in its multiplicative group.