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Algorithms determining finite simple images of finitely presented groups. (English) Zbl 1446.20051
According to a result of M. R. Bridson and H. Wilton [Invent. Math. 202, No. 2, 839–874 (2015; Zbl 1360.20020)], there is no algorithm that, given a finite presentation, determines whether or not the group presented has a nontrivial finite quotient. Consequently there is no algorithm that determines the finite simple images of a finitely presented group. On the other hand, by [W. Plesken, A. Fabiańska J. Algebra 322, 914–935 (2009; Zbl 1253.20033)] and [S. Jambor J. Algebra 423, 1109–1142 (2015; Zbl 1315.20034)], there is an algorithm that determines the images of a finitely presented group that are isomorphic to \(\mathrm{PSL}(2,q)\) or \(\mathrm{PGL}(2,q)\).
The article under review is dedicated to the study of the following question: for which collections of finite simple groups is there an algorithm that determines the members of the collection that are images of an arbitrary finitely presented group \(G\)?
For groups of unbounded rank, the authors get a negative answer. In the following statement, the dimension of a classical group is the dimension of its natural module.
Theorem 1. Let \(\mathcal{F}\) be a set of finite simple groups that either contains an infinite number of alternating groups, or contains a classical group of dimension \(n\) for infinitely many values of \(n\). Then there is no algorithm that, given a finite presentation, does either of the following: (i) determines whether or not the group presented has at least one quotient in \(\mathcal{F}\); (ii) determines whether or not the group presented has infinitely many quotients in \(\mathcal{F}\).
Let \(X\) be a fixed untwisted Lie type, and for a prime power \(q\) let \(X (q)\) be the simple group of type \(X\) over \(\mathbb{F}_{q}\). If \(X(q)\) possesses a graph automorphism of order \(d \in \{2, 3\}\), let \(^{d}X (q)\) be the corresponding twisted simple group over \(\mathbb{F}_{q}\). Let \(\mathrm{ID}( ^{d}X(q))\) be the group generated by inner and diagonal automorphisms of \(^{d}X(q)\). For a fixed Lie type \(X\) let \(\mathcal{X}^{1}=\{ X(q) \mid \text{ all prime powers } q \}\) and \(\mathcal{X}^{d}=\{ G \mid \, ^{d}X(q) \leq G \leq \mathrm{ID}(^{d}X(q)) \mbox{ for some } q\}\) if \(d\in \{2,3\}\).
Theorem 2. Let \(X\) be a fixed Lie type, and let \(\mathcal{X}^{d}\) be the corresponding class of groups (\(d \in \{1,2,3\}\)). Then there is an algorithm that determines whether or not any given finitely presented group has infinitely many quotients in \(\mathcal{X}^{d}\); moreover, if there are finitely many such quotients, the algorithm determines them. In particular, the algorithm determines whether or not any given finitely presented group has at least one quotient in \(\mathcal{X}^{d}\).

20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20D06 Simple groups: alternating groups and groups of Lie type
20F05 Generators, relations, and presentations of groups
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