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Squares of real conjugacy classes in finite groups

Antonio Beltrán, María José Felipe, Carmen Melchor

Abstract

We prove that if a finite group $G$ contains a conjugacy class $K$ whose square is of the form $1 \cup D$, where $D$ is a conjugacy class of $G$, then $\langle K \rangle$ is a solvable proper normal subgroup of $G$ and we completely determine its structure. We also obtain the structure of those groups in which the assumption above is true for all non-central conjugacy classes and when every conjugacy class satisfies that its square is the union of all central conjugacy classes except at most one.

Squares of real conjugacy classes in finite groups

Abstract

We prove that if a finite group contains a conjugacy class whose square is of the form , where is a conjugacy class of , then is a solvable proper normal subgroup of and we completely determine its structure. We also obtain the structure of those groups in which the assumption above is true for all non-central conjugacy classes and when every conjugacy class satisfies that its square is the union of all central conjugacy classes except at most one.
Paper Structure (3 sections, 7 theorems, 22 equations)

This paper contains 3 sections, 7 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Proofs

Key Result

theorem 1

Let $G$ be a finite group. Suppose that $P\in$ Syl$_{2}(G)$ and $j\in P$ such that $j^2=1\neq j$ and $P\cap \lbrace j^g \vert g\in G\rbrace=\lbrace j \rbrace$. Then O$_{2'}(G)\langle j \rangle \unlhd G$.

Theorems & Definitions (16)

  • theorem 1
  • lemma 1
  • proof
  • lemma 2: Lemma 3 of BerkoKaza
  • theorem 2: Theorem A of GuralnickNavarro
  • lemma 3: Theorem 1.2.6 of Michler
  • lemma 4
  • proof
  • proof : of Theorem A
  • lemma 5
  • ...and 6 more