The Isomorphism Problem for Higman-Thompson groups

TL;DR

This paper establishes a precise criterion for when Higman-Thompson groups are isomorphic, based on parameters n, r, m, s, and their greatest common divisors, advancing understanding of their algebraic structure.

Contribution

The paper provides a complete characterization of isomorphisms between Higman-Thompson groups using gcd conditions, resolving a fundamental question in their classification.

Findings

Higman-Thompson groups are isomorphic iff m=n and gcd(n-1,r)=gcd(n-1,s)

The result clarifies the algebraic structure and classification of these groups

Provides a definitive criterion for group isomorphism in this class

Abstract

We prove that the Higman-Thompson groups $G_{n,r}^+$ and $G_{m,s}^+$ are isomorphic if and only if $m=n$ and $\mbox{gcd}(n-1,r)=\mbox{gcd}(n-1,s)$.