Algorithms determining finite simple images of finitely presented groups

TL;DR

This paper investigates the existence of algorithms to determine finite simple images of finitely presented groups, proving both the possibility and impossibility depending on the types of simple groups involved.

Contribution

It establishes new results on which classes of finite simple groups admit such algorithms, highlighting positive results for bounded rank Lie type groups and negative results for others.

Findings

No algorithm exists for collections containing infinitely many alternating groups.

Algorithms exist for fixed Lie type groups of bounded rank.

The paper characterizes when finitely presented groups have finitely or infinitely many simple images.

Abstract

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for families of simple groups of Lie type of bounded rank, we obtain positive results. For example, given any fixed untwisted Lie type $X$ there is an algorithm that determines whether or not an arbitrary finitely presented group has infinitely many simple images isomorphic to $X(q)$ for some $q$, and if there are finitely many, the algorithm determines them.