Group theory
Large subgroups in finite groups
Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 253-257.

Following Isaacs (see [6, p. 94]), we call a normal subgroup N of a finite group G large, if CG(N)N, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing large subgroups in finite groups (see Theorems A and C). We also consider the more specialised problems of finding large (non-abelian) nilpotent as well as abelian subgroups in soluble groups.

Suivant la terminologie introduite par Isaacs (voir [6], p. 94), nous disons qu'un sous-groupe distingué N d'un groupe fini G est grand si CG(N)N, de sorte que N est d'indice borné dans G. Notre but principal est d'établir des résultats permettant de produire de façon systématique des grands sous-groupes dans les groupes finis (voir les théorèmes A et C). Nous considérons également les problèmes plus particuliers qui se posent pour trouver de grands sous-groupes nilpotents (non commutatifs) ainsi que de grands sous-groupes commutatifs dans les groupes résolubles.

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Published online:
DOI: 10.1016/j.crma.2018.01.020
Aivazidis, Stefanos 1; Müller, Thomas W. 2

1 Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 9190401, Israel
2 School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, London E1 4NS, United Kingdom
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Aivazidis, Stefanos; Müller, Thomas W. Large subgroups in finite groups. Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 253-257. doi : 10.1016/j.crma.2018.01.020. http://www.numdam.org/articles/10.1016/j.crma.2018.01.020/

[1] Aivazidis, S.; Müller, T. On residuals of finite groups, 2017 | arXiv

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[3] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.8, 2017.

[4] Gaschütz, W. Über die Φ-Untergruppe endlicher Gruppen, Math. Z., Volume 58 (1953), pp. 160-170

[5] Hall, P.; Higman, G. On the p-length of p-soluble groups and reduction theorems for Burnside's problem, Proc. London Math. Soc. (3), Volume 6 (1956), pp. 1-42

[6] Isaacs, I.M. Finite Group Theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, Providence, RI, USA, 2008

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