On the relation of character codegrees and the minimal faithful quasi-permutation representation degree of p-groups

Abstract

For a finite group G, we denote by c ( G ) , the minimal degree of a faithful representation of G by quasi-permutation matrices over C . For an irreducible character χ of G, the codegree of χ is defined as cod ( χ ) = | G / ker ( χ ) | / χ ( 1 ) . In this article, we establish equality between c ( G ) and a Q ≥ 0 -sum of codegrees of some irreducible characters of a non-abelian p-group G of odd order. We also study the relation between c ( G ) and irreducible character codegrees for various classes of non-abelian p-groups, such as, p-groups with cyclic center, maximal class p-groups, GVZ p-groups, and others.