The perfect groups of order up to two million

Hulpke, Alexander

doi:10.1090/mcom/3684

The perfect groups of order up to two million
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by Alexander Hulpke;

Math. Comp. 91 (2022), 1007-1017

DOI: https://doi.org/10.1090/mcom/3684

Published electronically: September 24, 2021

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Abstract:

We enumerate the $15768$ perfect groups of order up to $2\cdot 10^6$, up to isomorphism, thus also completing the missing cases reported by Holt and Plesken [Perfect groups, The Clarendon Press, Oxford University Press, New York, 1989]. The work supplements the by now well-understood computer classifications of solvable groups, illustrating scope and feasibility of the enumeration process for nonsolvable groups.

References

Hans Ulrich Besche, Bettina Eick, and E. A. O’Brien, A millennium project: constructing small groups, Internat. J. Algebra Comput. 12 (2002), no. 5, 623–644. MR 1935567, DOI 10.1142/S0218196702001115
John J. Cannon, Bruce C. Cox, and Derek F. Holt, Computing the subgroups of a permutation group, J. Symbolic Comput. 31 (2001), no. 1-2, 149–161. Computational algebra and number theory (Milwaukee, WI, 1996). MR 1806212, DOI 10.1006/jsco.2000.1012
John J. Cannon and Derek F. Holt, Automorphism group computation and isomorphism testing in finite groups, J. Symbolic Comput. 35 (2003), no. 3, 241–267. MR 1962794, DOI 10.1016/S0747-7171(02)00133-5
John J. Cannon, Derek F. Holt, Michael Slattery, and Allan K. Steel, Computing subgroups of bounded index in a finite group, J. Symbolic Comput. 40 (2005), no. 2, 1013–1022. MR 2167681, DOI 10.1016/j.jsc.2005.02.002
Heiko Dietrich and Alexander Hulpke, Universal covers of finite groups, J. Algebra 569 (2021), 681–712. MR 4187252, DOI 10.1016/j.jalgebra.2020.10.032
Bettina Eick, Max Horn, and Alexander Hulpke, Constructing groups of ‘small’ order: recent results and open problems, Algorithmic and experimental methods in algebra, geometry, and number theory, Springer, Cham, 2017, pp. 199–211. MR 3792726
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11, http://www.gap-system.org, 2020.
D. F. Holt, Enumerating perfect groups, J. London Math. Soc. (2) 39 (1989), no. 1, 67–78. MR 989920, DOI 10.1112/jlms/s2-39.1.67
Derek F. Holt, Bettina Eick, and Eamonn A. O’Brien, Handbook of computational group theory, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2129747, DOI 10.1201/9781420035216
Derek F. Holt and W. Plesken, Perfect groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. With an appendix by W. Hanrath; Oxford Science Publications. MR 1025760
Alexander Hulpke, Representing subgroups of finitely presented groups by quotient subgroups, Experiment. Math. 10 (2001), no. 3, 369–381. MR 1917425, DOI 10.1080/10586458.2001.10504457
Alexander Hulpke, Calculation of the subgroups of a trivial-fitting group, ISSAC 2013—Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2013, pp. 205–210. MR 3206359, DOI 10.1145/2465506.2465525
Joachim Neubüser, Untersuchungen des Untergruppenverbandes endlicher Gruppen auf einer programmgesteuerten elektronischen Dualmaschine, Numer. Math. 2 (1960), 280–292., DOI 10.1007/BF01386229
D. J. S. Robinson, Applications of cohomology to the theory of groups, Groups—St. Andrews 1981 (St. Andrews, 1981) London Math. Soc. Lecture Note Ser., vol. 71, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 46–80. MR 679154
Jack Schmidt, Finite groups have short rewriting systems, Computational group theory and the theory of groups, II, Contemp. Math., vol. 511, Amer. Math. Soc., Providence, RI, 2010, pp. 185–200. MR 2655300, DOI 10.1090/conm/511/10049
Charles C. Sims, Computation with finitely presented groups, Encyclopedia of Mathematics and its Applications, vol. 48, Cambridge University Press, Cambridge, 1994. MR 1267733, DOI 10.1017/CBO9780511574702
Michael J. Smith, Computing automorphisms of finite soluble groups, Ph.D. Thesis, Australian National University, 1994.