Abstract
Let G be a finite group. Goldschmidt, Flores, and Foote investigated the concept: Let K ≤ G. A subgroup H of K is called strongly closed in K with respect to G if H g ∩ K ≤ H for all g ∈ G. In particular, when H is a subgroup of prime-power order and K is a Sylow subgroup containing it, H is simply said to be a strongly closed subgroup. Bianchi and the others called a subgroup H of G an H-subgroup if N G (H) ∩ H g ≤ H for all g ∈ G. In fact, an H-subgroup of prime power order is the same as a strongly closed subgroup. We give the characterizations of finite non-T-groups whose maximal subgroups of even order are solvable T-groups by H-subgroups or strongly closed subgroups. Moreover, the structure of finite non-T-groups whose maximal subgroups of even order are solvable T-groups may be difficult to give if we do not use normality.
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References
Aschbacher M., Finite Group Theory, Cambridge Univ. Press, Cambridge (2000).
Doerk K. and Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin and New York (1992).
Huppert B., Endliche Gruppen. I, Springer-Verlag, Berlin (1968).
Robinson D. J. S., A Course in the Theory of Groups, Springer-Verlag, Berlin, Heidelberg, and New York (1982).
Goldschmidt D., “Strongly closed 2-subgroups of finite groups,” in: Proc. Conf. on Finite Groups, Univ. Utah, Park City, Utah, 1975, pp. 109–110.
Goldschmidt D. M., “2-Fusion in finite groups,” Ann. Math., 99, 70–117 (1974).
Goldschmidt D., “Strongly closed 2-subgroups of finite groups,” Ann. Math., 102, 475–489 (1975).
Flores R. J. and Foote R. M., “Strongly closed subgroups of finite groups,” Adv. Math., 222, 453–484 (2009).
Robinson D. J. S., “A note on finite groups in which normality is transitive,” Proc. Amer. Math. Soc., 19, 933–937 (1968).
Robinson D. J. S., “Groups in which normality is a transitive relation,” Proc. Cambridge Philos. Soc., 60, 21–38 (1964).
Robinson D. J. S., “Groups which are minimal with respect to normality being transitive,” Pacific. J.Math., 31, 777–785 (1969).
Bianchi M., Mauri A. G. B., Herzog M., and Verardi L., “On finite solvable groups in which normality is a transitive relation,” J. Group Theory, 3, 147–156 (2000).
Asaad M., “On p-nilpotence and supersolvability of finite groups,” Comm. Algebra, 34, No. 1, 189–195 (2006).
Ballester-Bolinches A., Esteban-Romero R., and Asaad M., Products of Finite Groups, De Gruyter, Berlin (2010).
Ballester-Bolinches A. and Esteban-Romero R., “On finite T-groups,” J. Austral. Math. Soc., 75, 181–191 (2003).
Beidleman J. C. and Heineken H., “Finite minimal non-T 1-groups,” J. Algebra, 319, 1685–1695 (2008).
Csörgo P. and Herzog M., “On supersolvable groups and the nilpotator,” Comm. Algebra, 32, No. 2, 609–620 (2004).
Ramadan M., “The influence of certain Abelian subgroups on the structure of finite groups,” Algebra Colloq., 3, 479–484 (2008).
Guo X.Y. and Wei X. B., “The influence of H -subgroups on the structure of finite groups,” J. Group Theory, 13, 267–276 (2009).
Shen Z.C. and Li S. R., “Finite groups with H -subgroups or strongly closed subgroups,” J.Group Theory, 15, 85–100 (2012).
Shen Z. C. and Du N., “Finite groups with H-subgroups,” Algebra Colloq., 20, No. 3, 421–426 (2013).
Shen Z. C. and Shi W. J., “Finite groups with some pronormal subgroups,” Acta Math. Sin., 27, 715–724 (2011).
Shen Z. C., Liu W., and Kong X. H., “Finite groups with self-conjugate-permutable subgroups,” Comm. Algebra, 38, 1715–1724 (2012).
Gorenstein D., Finite Groups, Chelsea, New York (1968).
Li S. R., “On minimal non-PE-groups,” J. Pure Appl. Algebra, 132, 149–158 (1998).
Li Y. M., “Finite groups with NE-subgroups,” J. Group Theory, 9, 49–58 (2006).
Thompson J. G., “Nonsolvable finite groups all of whose local subgroups are solvable. I,” Bull. Amer. Math. Soc., 74, 383–437 (1968).
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Original Russian Text Copyright © 2014 Shen Zh.C., Shi W.J., and Shen R.L.
The authors were supported by the Natural Science Foundation of China (Grants 11171364 and 11301532), the CPSF (Grant 2012T50010), and the Innovation Foundation of Chongqing (KJTD201321). The first author and corresponding author made the same contribution to this paper.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 3, pp. 706–714, May–June, 2014.
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Shen, Z.C., Shi, W.J. & Shen, R.L. On the strongly closed subgroups or H-subgroups of finite groups. Sib Math J 55, 578–584 (2014). https://doi.org/10.1134/S0037446614030197
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DOI: https://doi.org/10.1134/S0037446614030197