On the Conjugacy Classes of Cyclic Non-normal Subgroups

Abstract

Let G be a finite p-group. Assume that $$\nu (G)$$ and $$\nu _c(G)$$ denote the number of conjugacy classes of non-normal subgroups and non-normal cyclic subgroups of G, respectively. In this paper, we completely classify the finite p-groups with $$\nu _c=p$$ or $$p+1$$ for an odd prime number p. Also, we classify the groups G with $$\nu (G)=\nu _c(G)=p^i, i\ge 1$$ .