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Powers of conjugacy classes in a finite group

Antonio Beltrán, Rachel Deborah Camina, María José Felipe, Carmen Melchor

Abstract

Many results have been established that show how the number of conjugacy classes appearing in the product of classes affect the structure of a finite group. The aim of this paper is to show several results about solvability concerning the case in which the power of a conjugacy class is a union of one or two conjugacy classes. Moreover, we show that the above conditions can be determined through the character table of the group.

Powers of conjugacy classes in a finite group

Abstract

Many results have been established that show how the number of conjugacy classes appearing in the product of classes affect the structure of a finite group. The aim of this paper is to show several results about solvability concerning the case in which the power of a conjugacy class is a union of one or two conjugacy classes. Moreover, we show that the above conditions can be determined through the character table of the group.
Paper Structure (4 sections, 20 theorems, 59 equations)

This paper contains 4 sections, 20 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Powers of classes which are classes
  3. Powers which are a union of the trivial class and another class
  4. Powers which are a union of a class and its inverse

Key Result

lemma 1

(Lemma 2.1 of GuralnickNavarro) Let $x\in G$, where $G$ is a finite group, and let $K=x^G$. Then the following are equivalent:

Theorems & Definitions (42)

  • lemma 1
  • theorem 1
  • proof
  • remark 1
  • corollary 1
  • proof
  • corollary 2
  • proof
  • remark 2
  • corollary 3
  • ...and 32 more