On the number of non-cyclic subgroups of finite p-groups

Abstract

Let $G$ be a finite $p$-group and $\delta(G)$ denote the number of all non-cyclic subgroups of $G$. In this paper, an upper bound for $\delta(G)$ is obtained. Furthermore, we prove that $\delta(G)\leq \delta(M_p(1, 1, 1) \times C_{p}^{n-3})$ (if $p=2$, then $\delta(G)\leq \delta(D_8\times C_{2}^{n-3})$), for any non-elementary abelian $p$-group $G$ of order $p^n$.