Abstract:
Let π is a set of prime numbers. A very broad generalization of notion of nilpotent group is the notion of π-decomposable group, i.e. the direct product of π-group and π′-group. In the paper, the description of the finite non-π-decomposable groups in which all 2-maximal subgroups are π-decomposable is obtained. The proof used the author's results connected with the notion of control the prime spectrum of finite simple groups. The finite nonnilpotent groups in which all 2-maximal subgroups are nilpotent was studied by Z. Janko in 1962 in case of nonsolvable groups and the author in 1968 in case of solvable groups.
Keywords:
finite group, simple group, π-decomposable group, maximal subgroup, control of prime spectrum of group.
Citation:
V. A. Belonogov, “Finite groups in which all 2-maximal subgroups are π-decomposable”, Trudy Inst. Mat. i Mekh. UrO RAN, 20, no. 2, 2014, 29–43; Proc. Steklov Inst. Math. (Suppl.), 289, suppl. 1 (2015), 26–41
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\by V.~A.~Belonogov
\paper Finite groups in which all $2$-maximal subgroups are $\pi$-decomposable
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2014
\vol 20
\issue 2
\pages 29--43
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\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2015
\vol 289
\issue , suppl. 1
\pages 26--41
\crossref{https://doi.org/10.1134/S008154381505003X}
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Linking options:
https://www.mathnet.ru/eng/timm1056
https://www.mathnet.ru/eng/timm/v20/i2/p29
This publication is cited in the following 11 articles:
Kh. Li, Ch. Van, I. N. Safonova, A. N. Skiba, “Konechnye gruppy s $\sigma$-abnormalnymi podgruppami Shmidta”, Sib. matem. zhurn., 64:3 (2023), 585–597
H. Li, Zh. Wang, I. N. Safonova, A. N. Skiba, “Finite Groups with $ \sigma $-Abnormal Schmidt Subgroups”, Sib Math J, 64:3 (2023), 629
V. N. Rizhik, I. N. Safonova, A. N. Skiba, “On the $\mathfrak{F}$-Norm of a Finite Group”, Proc. Steklov Inst. Math. (Suppl.), 317, suppl. 1 (2022), S136–S141
Zhigang Wang, Jin Guo, Inna N. Safonova, Alexander N. Skiba, “A Generalization of $\sigma $-Permutability”, Commun. Math. Stat., 10:3 (2022), 565
Viktoria S. Zakrevskaya, “Finite groups with generalized subnormal and generalized permutable subgroups”, Asian-European J. Math., 15:01 (2022)
Hu B., Huang J., Skiba A.N., “On the SIGMA-Nilpotent Norm and the SIGMA-Nilpotent Length of a Finite Group”, Glasg. Math. J., 63:1 (2021), PII S0017089520000051, 121–132
Hu B., Huang J., Skiba A.N., “Finite Groups With SIGMA-Frobenius Condition For Non-Normal SIGMA-Primary Subgroups”, J. Algebra. Appl., 19:3 (2020), 2050047
Zh. Chi, A. N. Skiba, “On semi-sigma-nilpotent finite groups”, J. Algebra. Appl., 18:10 (2019), 1950200
B. Hu, J. Huang, “On finite groups with generalized sigma-subnormal Schmidt subgroups”, Commun. Algebr., 46:7 (2018), 3127–3134
V. A. Kovaleva, “Konechnye gruppy s zadannymi obobschenno maksimalnymi podgruppami (obzor). II. Ot maksimalnykh tsepei k maksimalnym param”, PFMT, 2017, no. 2(31), 55–65
V. A. Belonogov, “Finite groups in which all maximal subgroups are $\pi$-closed. I”, Proc. Steklov Inst. Math. (Suppl.), 293, suppl. 1 (2016), 22–31
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