# Congruence condition on number of cyclic subgroups of small prime power order

Suppose $p$ is any prime number and $0 \le k \le 5$. Suppose $P$ is a group of prime power order where the prime is $p$. Then, the number of cyclic subgroups of $P$ of order $p^k$ is congruent to either $0$ or $1$ modulo $p$.
Note that the statement is trivially true for $p = 2$, so it suffices to prove it for odd $p$.