The number of conjugacy classes of nonnormal subgroups of finite p-groups

Abstract

Assume k is a positive integer and p is a prime. Let ν(G) be the number of conjugacy classes of nonnormal subgroups of a finite group G and NCN(p,k)={ν(G)∈[0,kp]|G is a finite p-group}. In this paper, the set NCN(p,k) is determined for k≤2, and it is discovered that there is a new gap in the values that ν(G) can take in the case of finite p-groups. In particular, the number of conjugacy classes of nonnormal subgroups of minimal non-abelian p-groups is determined.