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Every \(\mathrm{PSL}_2(13)\) in the Monster contains \(13A\)-elements. (English) Zbl 1339.20013

The Monster \(M\) is the largest of the 26 finite sporadic simple groups, and the only one whose maximal subgroups are not yet completely classified. Much work has however been done on this problem over the years (see for example the papers of P. E. Holmes and R. A. Wilson [J. Algebra 251, No. 1, 435-447 (2002; Zbl 1006.20015); J. Algebra 319, No. 7, 2653-2667 (2008; Zbl 1146.20009)], S. P. Norton [Lond. Math. Soc. Lect. Note Ser. 249, 198-214 (1998; Zbl 0908.20008)], S. P. Norton and the author [Proc. Lond. Math. Soc., III. Ser. 84, No. 3, 581-598 (2002; Zbl 1017.20009); J. Lond. Math. Soc., II. Ser. 87, No. 3, 943-962 (2013; Zbl 1281.20019)] and the author [LMS J. Comput. Math. 17, 33-46 (2014; Zbl 1297.20014)]), but a few obstinate cases remain.
One of these is the problem of classifying subgroups isomorphic to \(\mathrm{PSL}_2(13)\), whose normalizers might be maximal. Norton in the above-mentioned paper described some subgroups isomorphic to \(\mathrm{PSL}_2(13)\) and containing \(13A\)-elements of \(M\). No subgroup in \(M\) isomorphic to \(\mathrm{PSL}_2(13)\) and containing \(13B\)-elements is known. In the given paper, the author shows by conducting an exhaustive computational search that, in fact, no such subgroup exists. Notation of elements here follows the well-known “Atlas of finite groups” (1985; Zbl 0448.20013).

MSC:

20D08 Simple groups: sporadic groups
20E28 Maximal subgroups
20-04 Software, source code, etc. for problems pertaining to group theory
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References:

[1] DOI: 10.1016/j.jalgebra.2003.11.014 · Zbl 1146.20009 · doi:10.1016/j.jalgebra.2003.11.014
[2] DOI: 10.1112/S0024610702003976 · Zbl 1065.20026 · doi:10.1112/S0024610702003976
[3] DOI: 10.1006/jabr.2001.9037 · Zbl 1006.20015 · doi:10.1006/jabr.2001.9037
[4] Conway, An atlas of finite groups (1985)
[5] Norton, Proceedings of the Atlas Ten Years on Conference, Birmingham, 1995 pp 198– (1998)
[6] DOI: 10.1112/S1461157013000247 · Zbl 1297.20014 · doi:10.1112/S1461157013000247
[7] DOI: 10.1007/978-1-84800-988-2 · Zbl 1203.20012 · doi:10.1007/978-1-84800-988-2
[8] DOI: 10.1112/jlms/jds078 · Zbl 1281.20019 · doi:10.1112/jlms/jds078
[9] DOI: 10.1112/S0024611502013357 · Zbl 1017.20009 · doi:10.1112/S0024611502013357
[10] DOI: 10.1007/s000130050437 · Zbl 0956.20022 · doi:10.1007/s000130050437
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