Some classes of finite groups and mutually permutable products.
A group \(G=AB\) is said to be a `mutually permutable product' of its subgroups \(A\) and \(B\) if \(A\) permutes with every subgroup of \(B\) and \(B\) permutes with every subgroup of \(A\). This paper is devoted to the study of mutually permutable products of finite groups in relation with certain classes of groups. The authors consider the class \(\mathcal Y\) of finite groups satisfying a converse of Lagrange's theorem in the following sense: \(G\) is a `\(\mathcal Y\)-group' if for every subgroup \(H\) and all primes \(q\) dividing the index \(|G:H|\) there exists a subgroup \(K\) of \(G\) such that \(H\) is contained in \(K\) and \(|K:H|=q\). \(\mathcal Y\)-groups have been studied in [\textit{M. Weinstein} (ed.), Between nilpotent and solvable. Passaic, New Jersey: Polygonal Publishing House (1982; Zbl 0488.20001)] and it is proved, in particular, that they are supersoluble groups. Moreover, in the paper under review, the authors also consider finite groups whose chief factors are simple, the so called `SC-groups'. It is proved that the mutually permutable product of a \(\mathcal Y\)-group and an SC-group is an SC-group, by means of local versions, depending on prime numbers, of both classes of groups. A local characterization of \(\mathcal Y\)-groups was obtained by the last three authors [in J. Algebra 312, No. 1, 445-454 (2007; Zbl 1119.20019)]. -- Another result in the paper states that the product of pairwise mutually permutable \(\mathcal Y\)-groups is supersoluble. \textit{H. Heineken} and the third author proved [in J. Algebra 297, No. 2, 409-416 (2006; Zbl 1104.20024)] the following: If a finite group \(G\) is the mutually permutable product of subgroups \(A\) and \(B\) and \(G\) is a PST-group (that is, a group in which every subnormal subgroup permutes with all Sylow subgroups), then \(A\) and \(B\) are PST-groups. In the paper under review the authors present a local version of this result.
scientific article; zbMATH DE number 5300812
scientific article