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New characterizations for supersolvability of fusion system and $p$-nilpotency of finite groups

Shengmin Zhang, Zhencai Shen

Abstract

Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that $S_{i+1}/S_i$ is cyclic, $i=0,1,\cdots, n-1$, $S_i$ is strongly $\mathcal{F}$-closed, $i=0,1,\cdots,n$. In this paper, we investigate the characterizations for supersolvability of $\mathcal{F}_S (G)$ under the assumption that certain subgroups of $G$ satisfy different kinds of generalized normalities in section \ref{1003}. Moreover, we obtain the more advanced and remarkable result of characterizations for generalized saturated fusion system $\mathcal{F}$ in section \ref{Section 4}. Finally, we apply the results in section \ref{1003} and \ref{Section 4} and give characterizations for $p$-nilpotency of finite groups under the assumption that some subgroups of $G$ satisfy different kinds of generalized normalities.

New characterizations for supersolvability of fusion system and $p$-nilpotency of finite groups

Abstract

Let be a prime, be a -group and be a saturated fusion system over . Then is said to be supersolvable, if there exists a series of , namely , such that is cyclic, , is strongly -closed, . In this paper, we investigate the characterizations for supersolvability of under the assumption that certain subgroups of satisfy different kinds of generalized normalities in section \ref{1003}. Moreover, we obtain the more advanced and remarkable result of characterizations for generalized saturated fusion system in section \ref{Section 4}. Finally, we apply the results in section \ref{1003} and \ref{Section 4} and give characterizations for -nilpotency of finite groups under the assumption that some subgroups of satisfy different kinds of generalized normalities.
Paper Structure (5 sections, 81 theorems, 11 equations)

This paper contains 5 sections, 81 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterizations for supersolvability of $\mathcal{F}_S (G)$
  4. Characterizations for supersolvability of saturated fusion systems
  5. Applications about characterizations for $p$-nilpotency of finite groups

Key Result

Theorem 1.3

Let $p$ be a prime, $G$ be a finite group, and $S$ be a Sylow $p$-subgroup of $G$. Suppose that there exists a subgroup $D$ of $S$ with $1<|D|<|S|$ such that every subgroup of $S$ of order $|D|$ or $p|D|$ is abelian and weakly pronormal in $G$, then $\mathcal{F}_S (G)$ is supersolvable.

Theorems & Definitions (93)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: FJ
  • Definition 1.4
  • Definition 1.5
  • Example 1.6
  • Definition 1.7
  • Example 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 83 more