Let $G$ be a finite group whose order is divisible by the prime $p$ . Suppose $p^m$ is the highest power of $p$ which is a factor of $|G|$ and set $$k = \frac{|G|}{p^m}.$$ Then

1. the group $G$ contains at least one subgroup of order $p^m$ ,
2. any two subgroups of $G$ of order $p^m$ are conjugate, and
3. the number of subgroups of $G$ of order $p^m$ is congruent to $1$ modulo $p$ and is a factor of $k$ .