Two classes of nonperiodic groups close to FC-groups (Q1071864)

The author characterizes groups having a normal infinite cyclic subgroup and satisfying either (1) each infinite cyclic subgroup has only finitely many conjugates or (2) each infinite finitely generated subgroup has only finitely many conjugates. A group \(G\) is in the first class if and only if \(G\) is an FC-group or \(G\) contains an FC-group \(H\) of index 2 such that \(H\) contains all elements of infinite order. \(G\) satisfies the second condition if and only if \(G\) is an FC-group or \(G\) is a non-central extension of an infinite cyclic group by a locally normal group.