Extreme Reidemeister spectra of finite groups

TL;DR

This work extends Reidemeister-theoretic notions to finite groups by defining the Reidemeister number $R(\varphi)$ through $\varphi$-twisted conjugacy and connecting it to fixed points of the induced action on conjugacy classes. It characterises finite groups with trivial Reidemeister spectrum as precisely those whose automorphisms are all class-preserving, and similarly identifies when the extended spectrum $\mathrm{ESpec}_R(G)$ is trivial, i.e. when endomorphisms are either class-preserving or fixed-point-free. The authors classify several families (e.g., groups with $\mathrm{Out}(G)=1$, certain simple and quasisimple groups) and provide extensive computational data for small groups, including the discovery of five finite groups with full extended Reidemeister spectrum up to order $<1536$. They also establish obstructions to full extended spectrum for broad group classes (odd order, nilpotent, quasisimple) and pose open questions about the existence of further examples, highlighting the nuanced interplay between automorphism structure and Reidemeister invariants in finite groups.

Abstract

We extend the notions of "$R_\infty$-property" and "full (extended) Reidemeister spectrum" to finite groups in a meaningful way. We provide examples of finite groups admitting these properties, if they exist, by looking at groups of small order as well as (quasi)simple groups.

Theorems & Definitions (25)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Theorem 2.4
• Corollary 2.5
• Corollary 2.6
• proof
• Proposition 2.7
• Definition 3.1
• Proposition 3.2