Lattice characterization of finite nilpotent groups

Abstract

The paper deals with subnormal and composition subgroups in the framework of weak congruence lattices of groups. Weak congruences of the composition subgroups of a group form a sublattice of the lattice of all weak congruences. We characterize normality and subnormality in purely lattice-theoretic terms. For a finite group G we prove: all subgroups of G are subnormal (i.e., G is nilpotent) if and only if the weak congruence lattice of G is lower semimodular.