Abstract:
Using the classification of finite simple groups, a description is obtained of finite groups having exactly three classes of conjugate maximal subgroups. If such a group is not solvable, then its factor group modulo its Frattini subgroup is isomorphic to PSL(2,7)PSL(2,7) or PSL(2,2p)PSL(2,2p), where pp is a prime. To prove this result, it was necessary to describe finite groups having at most two classes of conjugate nonnormal maximal subgroups.
Bibliography: 28 titles.
\Bibitem{Bel86}
\by V.~A.~Belonogov
\paper Finite groups with three classes of maximal subgroups
\jour Math. USSR-Sb.
\yr 1988
\vol 59
\issue 1
\pages 223--236
\mathnet{http://mi.mathnet.ru/eng/sm1919}
\crossref{https://doi.org/10.1070/SM1988v059n01ABEH003132}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=865936}
\zmath{https://zbmath.org/?q=an:0626.20007|0614.20004}
Linking options:
https://www.mathnet.ru/eng/sm1919
https://doi.org/10.1070/SM1988v059n01ABEH003132
https://www.mathnet.ru/eng/sm/v173/i2/p225
This publication is cited in the following 13 articles:
Wei Meng, Jiakuan Lu, “On the sum of non-cyclic subgroups order in a finite group”, Communications in Algebra, 52:3 (2024), 1084
Haowen Chen, Boru Zhang, Wei Meng, “On the sum of orders of non-cyclic and non-normal subgroups in a finite group”, International Electronic Journal of Algebra, 36:36 (2024), 206
Vyacheslav A. Belonogov, Viktor I. Zenkov, Anatoly S. Kondrat'ev, “Finite groups with four conjugacy classes of maximal subgroups”, European Journal of Mathematics, 9:4 (2023)
V. A. Belonogov, “Konechnye gruppy s chetyrmya klassami sopryazhennykh maksimalnykh podgrupp. III”, Tr. IMM UrO RAN, 27, no. 1, 2021, 5–18
Marcel Herzog, Patrizia Longobardi, Mercede Maj, “On a criterion for solvability of a finite group”, Communications in Algebra, 49:5 (2021), 2234
Changguo Shao, Antonio Beltrán, “Orbits of maximal invariant subgroups and solvability of finite groups”, Journal of Algebra, 539 (2019), 177
V. A. Belonogov, “Konechnye gruppy s chetyrmya klassami sopryazhennykh maksimalnykh podgrupp. II”, Sib. elektron. matem. izv., 15 (2018), 86–91
V. A. Belonogov, “Konechnye gruppy s chetyrmya klassami sopryazhennykh maksimalnykh podgrupp. I”, Tr. IMM UrO RAN, 23, no. 4, 2017, 52–62
V. A. Belonogov, “Konechnye gruppy, vse maksimalnye podgruppy kotorykh ππ-zamknuty. II”, Tr. IMM UrO RAN, 22, no. 3, 2016, 12–22
V. A. Belonogov, “Finite groups in which all 22-maximal subgroups are ππ-decomposable”, Proc. Steklov Inst. Math. (Suppl.), 289, suppl. 1 (2015), 26–41
Shi J.T., Zhang C., “Some Sufficient Conditions on the Number of Non-Abelian Subgroups of a Finite Group to Be Solvable”, Acta. Math. Sin.-English Ser., 27:5 (2011), 891–896
Shi J., Zhang C., “Finite Groups with Given Quantitative Non-Nilpotent Subgroups”, Commun. Algebr., 39:9 (2011), 3346–3355
Shi J., Shi W., Zhang C., “The Type of Conjugacy Classes of Maximal Subgroups and Characterization of Finite Groups”, Commun. Algebr., 38:1 (2010), 143–153
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