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On \(s\)-semipermutable maximal and minimal subgroups of Sylow \(p\)-subgroups of finite groups. (English) Zbl 1087.20015
Summary: A subgroup \(H\) of a group \(G\) is called semipermutable if it is permutable with every subgroup \(K\) of \(G\) with \((|H|,|K|)=1\). \(H\) is said to be \(s\)-semipermutable if it is permutable with every Sylow \(p\)-subgroup of \(G\) with \((p,|H|)=1\). In this article, we investigate the \(p\)-nilpotency of a group for which every maximal subgroup of its Sylow \(p\)-subgroups is \(s\)-semipermutable for some prime \(p\). We generalize some recent theorems by X. Guo and K. P. Shum [Arch. Math. 80, No. 6, 561-569 (2003; Zbl 1050.20010)].

MSC:
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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References:
[1] Chen Z., J. Math. China 18 pp 290– (1998)
[2] DOI: 10.1515/9783110870138
[3] Gorenstein D., Finite Groups (1968)
[4] DOI: 10.1007/s00013-003-0810-4 · Zbl 1050.20010
[5] Lennox J. C., Subnormal Subgroups of Groups (1987) · Zbl 0606.20001
[6] DOI: 10.1016/S0022-4049(99)00042-0 · Zbl 0967.20011
[7] DOI: 10.1007/BF00052096 · Zbl 0802.20019
[8] Robinson D. J. S., A Course in the Theory of Groups (1993)
[9] DOI: 10.1007/BF02761191 · Zbl 0437.20012
[10] DOI: 10.1016/0021-8693(64)90006-7 · Zbl 0119.26802
[11] DOI: 10.1006/jabr.1996.0103 · Zbl 0847.20010
[12] DOI: 10.1006/jabr.1999.8079 · Zbl 0953.20010
[13] Xu M., Introduction to the Theory of Finite Groups (1999)
[14] Zhang L., Acta Math. Sin. 29 pp 519– (1986)
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.