The following result is established.

THEOREM. Let G be a periodic, residually finite group with all subgroups sub-normal. Then G is nilpotent.

The well-known groups of Heineken and Mohamed [1] show that the hypothesis of residual finiteness cannot be omitted here, while examples in [5] show that a residually finite group with all subgroups subnormal need not be nilpotent. The proof of the Theorem will use the results of Möhres that a group with all subgroups subnormal is soluble [3] and that a periodic hypercentral group with all subgroups subnormal is nilpotent [4]. Borrowing an idea from [2], the plan is to construct certain subgroups H and K that intersect trivially, and to show that the subnormality of both leads to a contradiction.