×

zbMATH — the first resource for mathematics

\(SE\)-supplemented subgroups of finite groups. (English) Zbl 1287.20025
Let \(G\) be a finite group, \(H\) its subgroup, \(\text{Syl}(G)\) the set of all Sylow-subgroups. \(H\) is called S-quasinormal (S-quasinormally embedded) if \(HP=PH\) for all \(P\in\text{Syl}(G)\) (if every \(P\in\text{Syl}(H)\) is in \(\text{Syl}(S)\) for some S-quasinormal subgroup \(S\leq G\)). The subgroup \(H_s=\langle\bigcup\{S\leq H\mid S\text{ is S-quasinormally embedded in }G\}\rangle\) is called the SE-core of \(H\leq G\). \(H\) is called SE-supplemented if for some \(T\leq G\), \(HT=G\) and \(H\cap T\leq H_s\).
The question whether given a saturated formation \(\mathcal F\) containing all supersolvable groups, if for some normal subgroup \(E\leq G\), \(G/E\in\mathcal F\), under what conditions on \(E\), \(G\in\mathcal F\), has been investigated recently by many authors such as A. Ballester-Bolinches, Y. Wang and X. Guo [Glasg. Math. J. 42, No. 3, 383-389 (2000; Zbl 0968.20009)] and H. Wei, Y. Wang and Y. Li [Proc. Am. Math. Soc. 132, No. 8, 2197-2204 (2004; Zbl 1061.20014)].
The authors extending the results of these and other papers, answer the following questions affirmatively: Is it true that the already known results can be extended to Baer-local formations or strengthened by weakening conditions? Namely, the two main results are as follows: given a Baer-local formation \(\mathcal F\) containing all supersolvable groups and \(E\leq G\) a normal subgroup with \(G/E\in\mathcal F\), if all cyclic subgroups of prime order or of order 4 (all maximal subgroups of all \(P\in\text{Syl}(E)\)) are SE-supplemented then \(G\in\mathcal F\).

MSC:
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D40 Products of subgroups of abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D35 Subnormal subgroups of abstract finite groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. ASAAD, On maximal subgroups of finite group, Comm. Algebra 26 (1998), pp. 3647-3652. · Zbl 0915.20008
[2] M. ASAAD - P. CSOÈRGOÈ, Influence of minimal subgroups on the structure of finite group, Arch. Math. 72 (1999), pp. 401-404. · Zbl 0938.20013
[3] M. ASAAD - A. A. HELIEL, On S-quasinormally embedded subgroups of finite groups, J. Pure Appl. Algebra, 165 (2001), pp. 129-135. · Zbl 1011.20019
[4] A. BALLESTER-BOLINCHES - L. M. EZQUERRO, Classes of Finite Groups, Springer, Dordrecht, 2006. · Zbl 1102.20016
[5] A. BALLESTER-BOLINCHES - X. Y. GUO, On complemented subgroups of finite groups, Arch. Math. 72 (1999), pp. 161-166. · Zbl 0929.20015
[6] A. BALLESTER-BOLINCHES - M. C. PEDRAZA-AGUILERA, Sufficient conditions for supersolvability of finite groups, J. Pure Appl. Algebra, 127 (1998), pp. 113-118. · Zbl 0928.20020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.