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Some problems about products of conjugacy classes in finite groups

Antonio Beltrán, María José Felipe, Carmen Melchor

Abstract

We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved.

Some problems about products of conjugacy classes in finite groups

Abstract

We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved.
Paper Structure (5 sections, 20 theorems, 8 equations)

This paper contains 5 sections, 20 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Products of conjugacy classes: Arad and Herzog's conjecture
  3. Power of class which is a class
  4. Power of a class which is a union of the trivial class and another class
  5. Power of a class which is a union of a class and its inverse

Key Result

Theorem 2.2

(Preliminaries of MooriViet) Let $G$ be a group and let $a$, $b$, $c$$\in G$ be non-trivial elements of $G$. The following conditions are equivalent:

Theorems & Definitions (23)

  • Theorem 2.2
  • Theorem 2.4: Theorem 2.16 of Nuestro7
  • Theorem 2.5
  • Theorem 2.6: Theorem B of Nuestro6
  • Theorem 3.1: Theorem 2.2 of GuralnickNavarro
  • Theorem 3.2: Theorem 3.2 of GuralnickNavarro
  • Theorem 3.3: Theorem 2.2 of Nuestro7
  • Corollary 3.4: Corollary 2.4 of Nuestro7
  • Corollary 3.5: See Corollary 2.5 and Theorem 2.12 of Nuestro7
  • Corollary 3.6: Corollary 2.8 of Nuestro7
  • ...and 13 more