Co-degrees of irreducible characters in finite groups

Abstract

For a character χ of a finite group G, the number a(χ) := |G : ker χ |/χ (1) is called the co-degree of χ . The object of this paper is to study the connection between the structure of a finite group and the co-degrees of its irreducible characters.

A note on element orders and character codegrees

“…In this note, we try to find some connection between the element orders and the character codegrees in finite groups. The main result, which gives a positive answer to the Question B in [5], is as follows.…”

“…Let G be a finite group and let χ be a character of G. The positive integer a(χ) := |G : kerχ|/χ(1) is called, as in [5], the codegree of the character χ. In [4], we obtain a criterion for groups to be p-closed in terms of their character codegrees.…”

Codegrees and nilpotence class of p-groups

“…/ j 2 Irr.G/º. This definition of codegrees first appeared in [10] where the authors studied a graph associated with the set cod.G/. (The term co-degree of a character had earlier been used in [3] for a different quantity related to the character.…”

Finite Groups Whose Character Graphs Associated with Codegrees Have No Triangles

“…Wei define the character codegree graph ∆(G) of a finite group G in [8]. Specifically, the graph ∆(G) is defined as follows: the vertices of ∆(G) are the primes dividing the codegree of some nonprincipal irreducible character of G, and the vertices p and q are connected by an edge if and only if there exists a codegree of some nonprincipal irreducible character of G divisible by pq.…”

“…Specifically, the graph ∆(G) is defined as follows: the vertices of ∆(G) are the primes dividing the codegree of some nonprincipal irreducible character of G, and the vertices p and q are connected by an edge if and only if there exists a codegree of some nonprincipal irreducible character of G divisible by pq. In [8], the authors develop some properties of the graph ∆(G). For example, they show that if ∆(G) is connected, then its diameter is at most 3.…”