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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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References:
[1] Ax, J., The elementary theory of finite fields, Ann. Math., 88, 239-271, (1968) · Zbl 0195.05701
[2] Bray, J.N., Holt, D.F., Roney-Dougal, C.M.: The maximal subgroups of the low-dimensional finite classical groups. In: London Mathematical Society Lecture Note Series, vol. 407. Cambridge University Press, Cambridge (2013) · Zbl 1303.20053
[3] Bridson, MR; Wilton, H., The triviality problem for profinite completions, Invent. Math., 202, 839-874, (2015) · Zbl 1360.20020
[4] Carter, R.W.: Simple Groups of Lie Type. Wiley, London (1972) · Zbl 0248.20015
[5] Chatzidakis, Z.; den Dries, L.; Macintyre, A., Definable sets over finite fields, J. Reine Angew. Math., 427, 107-135, (1992) · Zbl 0759.11045
[6] Fried, MD; Haran, D.; Jarden, M., Effective counting of the points of definable sets over finite fields, Isr. J. Math., 85, 103-133, (1994) · Zbl 0826.11027
[7] Fried, M.D., Jarden, M.: Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete III 11. Springer, Heidelberg (1986)
[8] Fried, MD; Sacerdote, G., Solving diophantine problems over all residue class fields of a number field and all finite fields, Ann. Math., 104, 203-233, (1976) · Zbl 0376.02042
[9] Gorenstein, D., Lyons, R., Solomon, R.: The classification of the finite simple groups. Number 3. Part I. Chapter A. Almost simple \(K\)-groups. Mathematical Surveys and Monographs, 40.3. American Mathematical Society, Providence (1998) · Zbl 0890.20012
[10] Hrushovski, E.: The elementary theory of the Frobenius automorphisms. Preprint, version dated July 24, 2012. www.ma.huji.ac.il/ ehud/, 149 p
[11] Jambor, S., An \(L_2\)-quotient algorithm for finitely presented groups on arbitrarily many generators, J. Algebra, 423, 1109-1142, (2015) · Zbl 1315.20034
[12] Jambor, S.: An \(L_3-U_3\)-quotient algorithm for finitely presented groups. Ph.D. Thesis, RWTH Aachen University (2012)
[13] Larsen, M., Lubotzky, A.: Normal subgroup growth of linear groups: the \((G2, F4, E8)\)-theorem. In: Algebraic Groups and Arithmetic, pp. 441-468. Tata Inst. Fund. Res, Mumbai (2004) · Zbl 1161.20305
[14] Liebeck, MW; Macpherson, HD; Tent, K., Primitive permutation groups of bounded orbital diameter, Proc. Lond. Math. Soc., 100, 216-248, (2010) · Zbl 1225.20001
[15] Liebeck, MW; Seitz, GM, Subgroups generated by root elements in groups of Lie type, Ann. Math., 139, 293-361, (1994) · Zbl 0824.20041
[16] Liebeck, MW; Seitz, GM, On the subgroup structure of classical groups, Invent. Math., 134, 427-453, (1998) · Zbl 0920.20039
[17] Liebeck, MW; Seitz, GM, On the subgroup structure of exceptional groups of Lie type, Trans. Am. Math. Soc., 350, 3409-3482, (1998) · Zbl 0905.20031
[18] Liebeck, MW; Seitz, GM, Subgroups of exceptional algebraic groups which are irreducible on an adjoint or minimal module, J. Group Theory, 7, 347-372, (2004) · Zbl 1058.20037
[19] Liebeck, MW; Shalev, A., Residual properties of the modular group and other free products, J. Algebra, 268, 264-285, (2003) · Zbl 1034.20025
[20] Pink, R., Strong approximation for Zariski dense subgroups over arbitrary global fields, Comment. Math. Helv., 75, 608-643, (2000) · Zbl 0981.20036
[21] Platonov, V.P., Rapinchuk, A.S.: Algebraic Groups and Number Theory. Academic, San Diego (1994) · Zbl 0841.20046
[22] Plesken, W.; Fabiańska, A., An \(L_2\)-quotient algorithm for finitely presented groups, J. Algebra, 322, 914-935, (2009) · Zbl 1253.20033
[23] Rapinchuk, AS, On the congruence subgroup problem for algebraic groups, Sov. Math. Dokl., 39, 618-621, (1989) · Zbl 0692.20033
[24] Ryten, M.J.: Model theory of finite difference fields and simple groups. Ph.D. Thesis, University of Leeds (2007). http://www.maths.leeds.ac.uk/pure/staff/macpherson/ryten1.pdf
[25] Ryten, MJ; Tomašić, I., ACFA and measurability, Sel. Math. New Ser., 11, 523-537, (2005) · Zbl 1108.03043
[26] Tamburini, C.; Wilson, JS, A residual property of certain free products, Math. Z., 186, 525-530, (1984) · Zbl 0545.20019
[27] Tomanov, G., On the congruence-subgroup problem for some anisotropic algebraic groups over number fields, J. Reine Angew. Math., 402, 138-152, (1989) · Zbl 0673.20021
[28] Tomašić, I., Direct twisted Galois stratification, Ann. Pure Appl. Log., 169, 21-53, (2018) · Zbl 06803110
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