On \(p\)-decomposable groups (Q2760740)

Let \(\mathfrak F\) be the class of all \(p\)-decomposable groups, i.e. the groups \(G\) such that \(G=O_p(G)\times O_{p'}(G)\). A group \(G\) is \(\mathfrak F\)-constrained if \(C_G(G_{\mathfrak F})\leq G_{\mathfrak F}\), where \(G_{\mathfrak F}=O_p(G)\times O_{p'}(G)\). A subgroup \(I\) of \(G\) is an \(\mathfrak F\)-injector of \(G\) if \(I\cap N\) is an \(\mathfrak F\)-maximal subgroup of \(N\) for every subnormal subgroup \(N\) of \(G\). The authors describe the set of \(\mathfrak F\)-injectors of an \(\mathfrak F\)-constrained group. It is proved that a group with a unique conjugacy class of \(\mathfrak F\)-injectors is \(\mathfrak F\)-constrained. A characterisation of \(\mathfrak F\)-injectors in a \(p\)-solvable group is obtained. Properties of \(\mathfrak F\)-maximal and \(p\)-decomposable subgroups of \(p\)-solvable groups are analyzed.