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On the partial $Π$-property of second minimal or second maximal subgroups of Sylow subgroups of finite groups

Zhengtian Qiu, Jianjun Liu, Guiyun Chen

Abstract

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ Π$-property in $ G $ if if there exists a chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that for every $ G $-chief factor $ G_{i}/G_{i-1} $ $ (1\leq i\leq n) $ of $ \varGamma_{G} $, $ | G / G_{i-1} : N_{G/G_{i-1}} (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})| $ is a $ π(HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1}) $-number. In this paper, we study the influence of some second minimal or second maximal subgroups of a Sylow subgroup satisfying the partial $ Π$-property on the structure of a finite group.

On the partial $Π$-property of second minimal or second maximal subgroups of Sylow subgroups of finite groups

Abstract

Let be a subgroup of a finite group . We say that satisfies the partial -property in if if there exists a chief series of such that for every -chief factor of , is a -number. In this paper, we study the influence of some second minimal or second maximal subgroups of a Sylow subgroup satisfying the partial -property on the structure of a finite group.
Paper Structure (3 sections, 22 theorems, 10 equations)

This paper contains 3 sections, 22 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs

Key Result

Theorem 1.1

Let $E$ be a normal subgroup of $G$ and let $P$ be a Sylow $p$-subgroup of $E$. If every maximal subgroup of $P$ satisfies the partial $\Pi$-property in $G$, then either $E\leq Z_{{\mathcal{U}}_{p}}(G)$ or $|E|_{p}=p$.

Theorems & Definitions (35)

  • Theorem 1.1: Chen-2013
  • Theorem 1.2: Chen-2013
  • Theorem 1.3
  • Theorem 1.4
  • Example 1.5
  • Lemma 2.1: Chen-2013
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 25 more