Semipermutable \(\pi\)-subgroups.

*(English)*Zbl 1297.20018Author’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by W. Guo, K. P. Shum and A. N. Skiba [in J. Algebra 315, No. 1, 31-41 (2007; Zbl 1130.20017)] and the following unpublished result of Li: Let \(G=AB\) and let \(A\) contain some (specific) Sylow \(p\)-subgroup \(P\) of \(G\) as normal subgroup.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!

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by W. Guo, K. P. Shum and A. N. Skiba [in J. Algebra 315, No. 1, 31-41 (2007; Zbl 1130.20017)] and the following unpublished result of Li: Let \(G=AB\) and let \(A\) contain some (specific) Sylow \(p\)-subgroup \(P\) of \(G\) as normal subgroup.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!

Reviewer: Robert W. van der Waall (Huizen)

##### MSC:

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |

20D15 | Finite nilpotent groups, \(p\)-groups |

20D40 | Products of subgroups of abstract finite groups |

##### Keywords:

finite groups; semipermutable subgroups; permuting subgrups; normal closures; semipermutability; nilpotent groups; Sylow subgroups; products of subgroups; nilpotent complements
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##### 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 |

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