On Sylow permutable subgroups of finite groups. (English) Zbl 1380.20022
A subgroup \(H\) of a group \(G\) is called Sylow permutable, or \(S\)-permutable, in \(G\) if \(H\) permutes with all Sylow \(p\)-subgroups of \(G\) for all primes \(p\). A group \(G\) is said to be a PST-group if Sylow permutability is a transitive relation in \(G\). The authors of this interesting article proved that a group \(G\) which is factorised by a normal subgroup and a subnormal PST-subgroup of odd order is supersoluble (Theorem 1). The second main result is the following (Theorem 2) yields that the normal closure \(S^G\) of a subnormal PST-subgroup \(S\) of odd order of a group \(G\) is supersoluble, and the subgroup generated by subnormal PST-subgroups of \(G\) of odd order is supersoluble as well.
Reviewer: Igor Subbotin (Los Angeles)
MSC:
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D40 | Products of subgroups of abstract finite groups |
| 20D35 | Subnormal subgroups of abstract finite groups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
References:
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