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Bi-twisted conjugacy in finite groups

Pieter Senden, Sam Tertooy

Abstract

We provide two alternative ways to determine the number of (bi-)twisted conjugacy classes in a finite group: one by counting certain irreducible characters and one by counting certain twisted conjugacy classes of other endomorphisms. In addition, we show various inequalities and congruences for Reidemeister numbers, as well as relations between bi-twisted conjugacy, representation theory, and fixed-point free automorphisms.

Bi-twisted conjugacy in finite groups

Abstract

We provide two alternative ways to determine the number of (bi-)twisted conjugacy classes in a finite group: one by counting certain irreducible characters and one by counting certain twisted conjugacy classes of other endomorphisms. In addition, we show various inequalities and congruences for Reidemeister numbers, as well as relations between bi-twisted conjugacy, representation theory, and fixed-point free automorphisms.
Paper Structure (1 section, 4 theorems, 5 equations)

This paper contains 1 section, 4 theorems, 5 equations.

Table of Contents

  1. Introduction

Key Result

Proposition 1.1

Let $G$ be a finite group and $\psi \in \mathop{\mathrm{End}}\nolimits(G)$. Then $R(\psi)$ is equal to the number of conjugacy classes that satisfy $[g] = [\psi(g)]$.

Theorems & Definitions (5)

  • Proposition 1.1
  • Proposition 1.2: Tertooy25
  • proof
  • Theorem 1.3: Rowley95
  • Corollary 1.3