On the number of subgroups of given order in a finite $p$-group of exponent $p$

Abstract:

A. Kulakoff [1] showed that a noncyclic $p$-group of order ${p^m},p > 2$, contains $1 + p + k{p^2}$ subgroups of order ${p^n},0 < n < m$, where $k$ is a nonnegative integer. In this note we show that for $1 < n < m - 1$ a $p$-group of order ${p^m}$ and exponent $p$ contains $1 + p + 2{p^2} + k{p^3}$ subgroups of order ${p^n}$.