The number of conjugacy classes of noncyclic subgroups of finite nilpotent groups

Abstract

Let [Formula: see text] be a finite nilpotent group. We denote by [Formula: see text] the number of conjugacy classes of noncyclic subgroups of [Formula: see text], [Formula: see text] the number of conjugacy classes of subgroups of [Formula: see text]. In this paper, we give the formula for [Formula: see text], and prove that [Formula: see text] when the order of [Formula: see text] is prime power. Moreover, we get the minimum of [Formula: see text] according to [Formula: see text], where [Formula: see text] is the number of distinct noncyclic Sylow subgroups of [Formula: see text].