On groups with many almost normal subgroups (Q1913368)

In an FC group every finitely generated subgroup has finitely many conjugates. The authors define an anti-FC-group to be a group in which all non-finitely generated subgroups have only finitely many conjugates. Clearly centre-by-finite groups and groups satisfying the maximal condition are anti-FC. The authors describe the soluble anti-FC-groups not in these two classes. A soluble anti-FC-group is first shown to be an \(S_1\)-group and then different cases are considered. If \(G\) is a soluble anti-FC-group containing a \(p^\infty\)-subgroup \(P\) then different characterisations are given depending on whether \(P\) is central or not. In both cases, \(\text{Spec }G=\{p\}\). If \(G\) is an \(S_1\)-group without \(p^\infty\)-subgroups then its maximal normal periodic subgroup \(T\) is finite. Then \(G\) is anti-FC if and only if \(G/T\) is anti-FC. The description is then completed by considering soluble anti-FC-groups with no periodic normal subgroup.