Finite groups with gcd(χ(1), χc (1)) a prime

Abstract

The aim of this article is to study how the greatest common divisor of the degree and codegree of an irreducible character of a finite group influences its structure. We study a finite group GG with gcd(χ(1),χc(1)){\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1)) a prime for almost all irreducible characters χ\chi of GG, and obtain the following two conclusions: (1)There does not exist any finite group GG such that gcd(χ(1),χc(1)){\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1)) is a prime, for each χ∈Irr(G)♯\chi \in {\rm{Irr}}{\left(G)}^{\sharp }, where Irr(G)♯{\rm{Irr}}{\left(G)}^{\sharp } is the set of non-principal irreducible characters of GG.(2)Let GG be a finite group, if gcd(χ(1),χc(1)){\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1)) is a prime, for each χ∈Irr(G)\Lin(G)\chi \left\in {\rm{Irr}}\left(G)\backslash {\rm{Lin}}\left(G), then GG is solvable, where Lin(G){\rm{Lin}}\left(G) is the set of all linear irreducible characters of GG.

Keywords

• a finite group
• an irreducible character
• degree
• codegree
• the greatest common divisor
• structure
• 20d10
• 20d15
• 20d20