Finite groups with weakly \(s\)-semipermutable subgroups.
A subgroup \(H\) of a finite group \(G\) is said to be \(S\)-permutable in \(G\) if \(HS\) is a subgroup of \(G\) for all Sylow subgroups \(S\) of \(G\). This embedding property was introduced by Kegel, it has been extensively investigated and it turns out to be important in the structural study of the finite groups. An interesting variation of the \(S\)-permutability is the \(S\)-semipermutability which is defined as the permutability of a subgroup \(H\) with the Sylow subgroups corresponding to the primes not dividing its order. This subgroup embedding property has been considered in some recent papers dealing with problems concerning transitivity and subnormality. In this paper the author considers an interesting variation of \(S\)-semipermutability motivated by the property of c-supplementation introduced in the paper [\textit{A. Ballester-Bolinches, Y. Wang} and \textit{X. Guo}, Glasg. Math. J. 42, No. 3, 383-389 (2000; Zbl 0968.20009)]: a subgroup \(H\) of a finite group \(G\) is said to be weakly \(s\)-semipermutable in \(G\) if there exists a subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K\) is \(S\)-semipermutable in \(G\). This property is then used in order to derive information on the internal structure of a group \(G\) provided that various subgroups of \(G\) are weakly \(s\)-semipermutable in \(G\). The formation theory provides a wonderful framework to unify and generalise earlier results.
scientific article; zbMATH DE number 6005071