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 p -group of order p m , p > 2 {p^m},p > 2 , contains 1 + p + k p 2 1 + p + k{p^2} subgroups of order p n , 0 > n > m {p^n},0 > n > m , where k k is a nonnegative integer. In this note we show that for 1 > n > m − 1 1 > n > m - 1 a p p -group of order p m {p^m} and exponent p p contains 1 + p + 2 p 2 + k p 3 1 + p + 2{p^2} + k{p^3} subgroups of order p n {p^n} .