Subnormal, permutable, and embedded subgroups in finite groups. (Q657383)

A subgroup \(H\) of a group \(G\) is (a) permutable (S-permutable), if it is permutable with all subgroups (all Sylow subgroups) of \(G\), (b) semipermutable (S-semipermutable) if it is permutable with all subgroups (all Sylow subgroups) of coprime order of \(G\), (c) seminormal (S-seminormal), if it is normalized by all subgroups (all Sylow subgroups) of coprime order of \(G\). The purpose of this article is to study these subgroup embedding properties and to compare them with other permutability properties like those considered by \textit{A. Ballester-Bolinches} and \textit{R. Esteban-Romero} [J. Algebra 251, No. 2, 727-738 (2002; Zbl 1010.20013)]. The following result (Theorem 1.5) is an example: For a solvable group \(G\) the following statements on all of its subnormal subgroups are equivalent: (i) they are seminomal, (ii) they are semipermutable, (iii) they are S-semipermutable, (iv) they are permutable.