Finite groups with submodular primary subgroups

We investigate groups with submodular primary subgroups. We establish that a primary subgroup H of a soluble group G is submodular if and only if H is U 1 -subnormal in G . Here U 1 is the formation of all supersoluble groups of square-free exponents. We describe the structure of a group with submodular cyclic primary subgroups. It is proved that in a group with nilpotent derived subgroup, submodularity of cyclic primary subgroups is equivalent to submodularity of Sylow subgroups.