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ON WEAKLY s-PERMUTABLY EMBEDDED SUBGROUPS OF FINITE GROUPS II

Part of: Representation theory of groups

Published online by Cambridge University Press:  21 May 2012

SHOUHONG QIAO  and
YANMING WANG
SHOUHONG QIAO
Affiliation:
School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, PR China (email: [email protected])
YANMING WANG*
Affiliation:
Lingnan College and Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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A subgroup H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup Hse of G contained in H such that G=HT and H∩T≤Hse. In this note, we study the influence of the weakly s-permutably embedded property of subgroups on the structure of G, and obtain the following theorem. Let ℱ be a saturated formation containing 𝒰, the class of all supersolvable groups, and G a group with E as a normal subgroup of G such that G/E∈ℱ. Suppose that P has a subgroup D such that 1<∣D∣<∣P∣ and all subgroups H of P with order ∣H∣=∣D∣ are s-permutably embedded in G. Also, when p=2 and ∣D∣=2 , we suppose that each cyclic subgroup of P of order four is weakly s-permutably embedded in G. Then G∈ℱ.

Keywords

weakly s-permutably embedded subgroupthe generalised fitting subgroupp-nilpotent groupsaturated formation

MSC classification

Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks 20D15: Nilpotent groups, $p$-groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 1 , August 2012 , pp. 41 - 49
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

Project supported by NSF China 11171353.

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