×

zbMATH — the first resource for mathematics

Semipermutable \(\pi\)-subgroups. (English) Zbl 1297.20018
Author’s summary: “Let \(H\) be a \(\pi\)-subgroup of \(G\), and assume that \(HQ=QH\) for every Sylow \(q\)-subgroup \(Q\) of \(G\) for all primes \(q\) not dividing \(|H|\). We show that the normal closure \(H^G\) of \(H\) in \(G\) has a nilpotent \(\pi\)-complement, and in the case where \(\pi\) consists of just one prime, \(H^G\) is solvable.”
There are several theorems and corollaries proved in this paper, thereby extending and generalizing results of others in passing, such as those obtained by W. Guo, K. P. Shum and A. N. Skiba [in J. Algebra 315, No. 1, 31-41 (2007; Zbl 1130.20017)] and the following unpublished result of Li: Let \(G=AB\) and let \(A\) contain some (specific) Sylow \(p\)-subgroup \(P\) of \(G\) as normal subgroup. Then, if \(P\) permutes with all Sylow \(q\)-subgroups of \(B\) with \(q\neq p\), \(G\) is \(p\)-solvable.
We omit giving an overview of the obtained results of the paper. They certainly enhance the existing literature. Highly recommended!

MSC:
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D15 Finite nilpotent groups, \(p\)-groups
20D40 Products of subgroups of abstract finite groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Guo, W.; Shum, K.P.; Skiba, X-semipermutable subgroups of finite groups, J. of Algebra, 315, 31-41, (2007) · Zbl 1130.20017
[2] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967. · Zbl 0217.07201
[3] I. M. Isaacs, Finite group theory, GSM 92, American Math. Soc., Providence RI (2008). · Zbl 1169.20001
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.