Residually finite FC-groups (Q1081697)

The question of when a residually finite periodic FC-group can be embedded in a direct product of finite groups has received considerable attention. It is known to be the case if G/G' has finite exponent and the author shows (Theorem 2) that it is also the case if G/G' is countable. Theorem 3 claims to give a counterexample with G/G' a direct product of cyclic groups. Unfortunately, the author's description of the group B at the beginning of the construction is incorrect. The group B is intended to be Dieudonné's example of a residually finite abelian group such that \(B/\Omega_ 1(B)\) is a direct product of cyclic groups but B itself is not. It is not clear to the reviewer that, if the necessary changes to the description of B are made, the remainder of the proof can be adapted. A second question is whether a residually finite periodic FC-group with countable centre can be embedded in a direct product of finite groups. Theorem 4 gives a step towards this, showing that if G is also metabelian then it can be so embedded.