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A criterion for a normal subgroup to be hypercentral based on class sizes

Antonio Beltrán

Abstract

Let $G$ be a finite group and $N$ a normal subgroup of $G$. We prove that the knowledge of the sizes of the conjugacy classes of $G$ that are contained in $N$ and of their multiplicities provides information of $N$ in relation to the structure of $G$. Among other results, we obtain a criterion to determine whether a Sylow $p$-subgroup of $N$ lies in the hypercentre of $G$ for a fixed prime $p$, and therefore, whether the whole subgroup $N$ is hypercentral in $G$.

A criterion for a normal subgroup to be hypercentral based on class sizes

Abstract

Let be a finite group and a normal subgroup of . We prove that the knowledge of the sizes of the conjugacy classes of that are contained in and of their multiplicities provides information of in relation to the structure of . Among other results, we obtain a criterion to determine whether a Sylow -subgroup of lies in the hypercentre of for a fixed prime , and therefore, whether the whole subgroup is hypercentral in .
Paper Structure (3 sections, 5 theorems, 17 equations)

This paper contains 3 sections, 5 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Proofs

Key Result

Lemma 2.1

Let $G$ be a finite group. A normal $p$-subgroup $N$ of $G$ lies in ${\bf Z}_\infty(G)$ if and only if $G/{\bf C}_G(N)$ is a $p$-group.

Theorems & Definitions (9)

  • Lemma 2.1: p. 739, Huppert2
  • Lemma 2.2: Lemma 2, CHM
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • proof : Proof of Corollary B
  • Theorem 3.2
  • proof