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On the sharp Baer-Suzuki theorem for the \(\pi \)-radical: sporadic groups. (English. Russian original) Zbl 07505252

Sib. Math. J. 63, No. 2, 387-394 (2022); translation from Sib. Mat. Zh. 63, No. 2, 465-473 (2022).
Summary: Let \(\pi\) be a proper subset of the set of all primes and \({|\pi|\geq 2} \). Denote the smallest prime not in \(\pi\) by \(r\) and let \(m=r\) if \(r=2,3 \), and \(m=r-1\) if \(r\geq 5 \). We study the following conjecture: A conjugacy class \(D\) of a finite group \(G\) lies in the \(\pi \)-radical \(\text{O}_{\pi}(G)\) of \(G\) if and only if every \(m\) elements of \(D\) generate a \(\pi \)-subgroup. We confirm this conjecture for the groups \(G\) whose every nonabelian composition factor is isomorphic to a sporadic or alternating group.

MSC:

20Dxx Abstract finite groups
20Fxx Special aspects of infinite or finite groups
20Exx Structure and classification of infinite or finite groups

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References:

[1] Baer, R., Engelsche Elemente Noetherscher Gruppen, Math. Ann., 133, 256-270 (1957) · Zbl 0078.01501 · doi:10.1007/BF02547953
[2] Suzuki, M., Finite groups in which the centralizer of any element of order 2 is 2-closed, Ann. Math. (2), 82, 191-212 (1965) · Zbl 0132.01704 · doi:10.2307/1970569
[3] Alperin, J.; Lyons, R., On conjugacy classes of \(p \)-elements, J. Algebra, 19, 2, 536-537 (1971) · Zbl 0238.20026 · doi:10.1016/0021-8693(71)90086-X
[4] Mazurov, VD; Olshanskii, AY; Sozutov, AI, Infinite groups of finite period, Algebra Logic, 54, 2, 161-166 (2015) · Zbl 1332.20042 · doi:10.1007/s10469-015-9335-8
[5] Mamontov, AS, An analog of the Baer-Suzuki theorem for infinite groups, Sib. Math. J., 45, 2, 327-330 (2004) · Zbl 1079.20049 · doi:10.1023/B:SIMJ.0000021288.57892.c6
[6] Mamontov, AS, The Baer-Suzuki Theorem for groups of \(2 \)-exponent \(4 \), Algebra Logic, 53, 5, 422-424 (2014) · Zbl 1315.20039 · doi:10.1007/s10469-014-9302-9
[7] Palchik, EM, Generation of finite groups by pairs of conjugate elements, Dokl. NAN Belarusi, 55, 4, 19-20 (2011) · Zbl 1267.20027
[8] Revin, DO, On Baer-Suzuki \(\pi \)-theorems, Sib. Math. J., 52, 2, 340-347 (2011) · Zbl 1244.20016 · doi:10.1134/S0037446611020170
[9] Revin, DO, On a relation between the Sylow and Baer-Suzuki theorems, Sib. Math. J., 52, 5, 904-913 (2011) · Zbl 1237.20018 · doi:10.1134/S0037446611050156
[10] Sozutov, AI, On a generalization of the Baer-Suzuki theorem, Sib. Math. J., 45, 3, 561-562 (2000) · Zbl 0960.20016 · doi:10.1007/BF02674111
[11] Tyutyanov, VN, On the existence of solvable normal subgroups in finite groups, Math. Notes, 61, 5, 632-634 (1997) · Zbl 0929.20018 · doi:10.1007/BF02355085
[12] Tyutyanov, VN, A nonsimplicity criterion for a finite group, Izv. Francisk Skorina Gomel State University (Problems of Algebra), 16, 3, 125-137 (2000) · Zbl 1159.20323
[13] Flavell, P.; Guest, S.; Guralnick, R., Characterizations of the solvable radical, Proc. Amer. Math. Soc., 138, 4, 1161-1170 (2010) · Zbl 1202.20026 · doi:10.1090/S0002-9939-09-10066-7
[14] Gordeev, N.; Grunewald, F.; Kunyavskii, B.; Plotkin, E., A description of Baer-Suzuki type of the solvable radical of a finite group, J. Pure Appl. Algebra, 213, 2, 250-258 (2009) · Zbl 1163.20010 · doi:10.1016/j.jpaa.2008.06.006
[15] Gordeev, N.; Grunewald, F.; Kunyavskii, B.; Plotkin, E., Baer-Suzuki theorem for the solvable radical of a finite group, C. R. Acad. Sci. Paris, Ser. I, 347, 5-6, 217-222 (2009) · Zbl 1167.20012 · doi:10.1016/j.crma.2009.01.004
[16] Gordeev, N.; Grunewald, F.; Kunyavskii, B.; Plotkin, E., From Thompson to Baer-Suzuki: a sharp characterization of the solvable radical, J. Algebra, 323, 10, 2888-2904 (2010) · Zbl 1201.20008 · doi:10.1016/j.jalgebra.2010.01.032
[17] Guest, S., A solvable version of the Baer-Suzuki theorem, Trans. Amer. Math. Soc., 362, 5909-5946 (2010) · Zbl 1214.20025 · doi:10.1090/S0002-9947-2010-04932-3
[18] Guest, S.; Levy, D., Criteria for solvable radical membership via \(p \)-elements, J. Algebra, 415, 88-111 (2014) · Zbl 1298.20025 · doi:10.1016/j.jalgebra.2014.06.003
[19] Yang, N.; Revin, DO; Vdovin, EP, Baer-Suzuki theorem for the \(\pi \)-radical, Israel J. Math., 245, 1, 173-207 (2021) · Zbl 07456845 · doi:10.1007/s11856-021-2209-y
[20] Yang, N.; Wu, Z.; Revin, DO; Vdovin, EP, On the Sharp Baer-Suzuki Theorem for the \(\pi \)-Radical (2021) · Zbl 07456845
[21] Guralnick, R.; Saxl, J., Generation of finite almost simple groups by conjugates radical, J. Algebra, 268, 519-571 (2003) · Zbl 1037.20016 · doi:10.1016/S0021-8693(03)00182-0
[22] Di Martino, L.; Pellegrini, MA; Zalesski, AE, On generators and representations of the sporadic simple groups, Comm. Algebra, 42, 2, 880-908 (2014) · Zbl 1298.20043 · doi:10.1080/00927872.2012.729629
[23] Conway, JH; Curtis, RT; Norton, SP; Parker, RA; Wilson, RA, Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups (1985), Oxford: Clarendon, Oxford · Zbl 0568.20001
[24] Isaacs, IM, Character Theory of Finite Groups (1976), Providence: Chelsea, Providence · Zbl 0337.20005
[25] The GAP Group, GAP—Groups, Algorithms, and Programming (2018)
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