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

Abstract

In a recent article S\'ebastien 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 Kochend\"orffer's criterion for the existence of a faithful irreducible representation based on the structure of the socle of $G$.

Keywords

• faithful irreducible representation
• subgroup lattice
• boolean lattice