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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 $$\mathcal{X}^{d}$$.

##### MSC:
 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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