Groups with decomposable set of quasinormal subgroups (Q2752988)

A subgroup \(H\) of a group \(G\) is said to be quasinormal if \(HX=XH\) for all subgroups \(X\) of \(G\). The main result of the paper under review is a characterization of the groups for which the partially ordered set of quasinormal subgroups is decomposable into the direct product of two non-trivial partially ordered sets. This happens if and only if \(G=G_1\times G_2\) for some non-trivial subgroups \(G_1\) and \(G_2\) such that: (i) Every quasinormal subgroup of \(G_1\) and \(G_2\) is also quasinormal in \(G\); (ii) If \(H_1\) and \(H_2\) are quasinormal in \(G_1\) and \(G_2\), respectively, then the factor groups \(\ker(G_1:H_1)/H_1\) and \(\ker(G_2:H_2)/H_2\) have no elements of the same prime order. Here \(\ker(G:H)\), where \(H\) is a subgroup of the group \(G\) is the subgroup of all \(g\in G\) such that \(X^g=X\) for each subgroup \(X\) of \(G\) containing \(H\).