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 G G with gcd ( χ ( 1 ) , χ c ( 1 ) ) {\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1)) a prime for almost all irreducible characters χ \chi of G G , and obtain the following two conclusions: (1) There does not exist any finite group G G such that gcd ( χ ( 1 ) , χ c ( 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 G G . (2) Let G G be a finite group, if gcd ( χ ( 1 ) , χ c ( 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 G G is solvable, where Lin ( G ) {\rm{Lin}}\left(G) is the set of all linear irreducible characters of G G .