Torsionsgruppen, deren Untergruppen alle subnormal sind. (Torsion groups all subgroups of which are subnormal) (Q1121989)

It is well-known that in a nilpotent group all subgroups are subnormal, but the converse is false. Here it is shown that a group G, which is an extension of a nilpotent torsion group by a soluble group of finite exponent and has all its subgroups subnormal, is nilpotent. The proof of this is easily reduced to the case where G is a p-group, which is an extension of an abelian normal subgroup by an elementary abelian group.