Residually Finite Groups with all Subgroups Subnormal

The following result is established. T HEOREM . 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. 1991 Mathematics Subject Classification 20E15.