a lattice theoretic characterization for the existence of a faithful irreducible representation

In a recent article sébastien palcoux formulated a sufficient condition on the subgroup lattice of a finite group g that guarantees the existence of a faithful irreducible complex representation of g, and asked whether his condition is also necessary. in this short note we give an affirmative answer using kochendörffer’s criterion for the existence of a faithful irreducible representation based on the structure of the socle of g.