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

TL;DR

This paper establishes an upper bound for the number of non-cyclic subgroups in finite p-groups and characterizes the groups that attain this bound, focusing on non-elementary abelian p-groups.

Contribution

It provides a new upper bound for non-cyclic subgroups in finite p-groups and identifies the groups that maximize this count.

Findings

Upper bound for $\delta(G)$ in finite p-groups.

Equality cases characterized by specific group structures.

Applicable to non-elementary abelian p-groups of order $p^n$.

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 $p^n$.