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

Jia Liu, Li Ma, Wei Meng

Abstract

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

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

Abstract

Let be a finite -group and denote the number of all non-cyclic subgroups of . In this paper, an upper bound for is obtained. Furthermore, we prove that (if , then ), for any non-elementary abelian -group of order .
Paper Structure (3 sections, 16 theorems, 6 equations)

This paper contains 3 sections, 16 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Proof of Main Results

Key Result

Theorem 1.1

Let $G$ be a group of order $p^n$. Then and the equality holds if and only if $G$ is an elementary abelian $p$-group of order $p^n$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 6 more