On generalised FC-groups in which normality is a transitive relation. (Q2800020)

A group \(G\) is an FC-group, if all elements are contained in finite conjugacy classes. We call FC-groups \(\text{FC}^1\)-groups and define recursively \(\text{FC}^{n+1}\)-groups as groups all of whose elements \(x\) satisfy \(G/C_G(x)\in\text{FC}^n\). The set-theoretical union of all of these classes is the class of \(\text{FC}^*\)-groups. Denote by \(A_G(H)\) the intersection of all subgroups of \(G\) containing the subgroup \(H\) and call \(G\) an NNM-group if every non-normal subgroup of \(G\) is contained in a non-normal maximal subgroup of \(G\).NEWLINENEWLINE Results: If \(G\) is an \(\text{FC}^*\)-group with \(A_G(H)\) non-normal whenever \(H\) is non-normal in \(G\), then \(G\) is a T-group and NNM-group.NEWLINENEWLINE For \(\text{FC}^*\)-groups \(G\) the following are equivalent: (i) \(G\) is a soluble T-group, (ii) every subgroup of \(G\) is pronormal, (iii) every cyclic subgroup of \(G\) is pronormal, and \(G\) is then an FC-group.NEWLINENEWLINE For \(\text{FC}^*\)-groups \(G\) the following are equivalent: (i) \(G\) is a soluble T-group, (ii) all subgroups of \(G\) are NNM-groups. (Proposition 2.2, Theorem 2.3, Theorem 2.5). -- They are extensions of results by Kaplan and the second author.