Element orders and character codegrees

Abstract

For an irreducible character χ of a finite group G, the codegree of χ is defined by |G:kerχ|/χ(1). In this note, we show that if a finite solvable group G has an element of order m, then G admits an irreducible character of codegree divisible by m.

Codegrees of primitive characters of solvable groups

“…Here we mention a result of Qian ([10, Theorem 1.1]), which says that if a finite solvable group G has an element g of square-free order, then G must have an irreducible character of codegree divisible by the order o(g) of g. In a short time, Isaacs [5] established the same result for an arbitrary finite group. Recently Qian [11] strengthened further his previous result, showing that for every element g of a finite solvable group G, there exists necessarily some χ ∈ Irr(G) such that o(g) divides cod(χ).…”

“…The authors believe that among the above-mentioned results, the most interesting finding is the relation between the codegrees and the element orders (see [5,10,11] for example).…”

“…Motivated by the results in [5,10,11], Moretó considered the converse relation of codegrees and element orders and proposed an interesting question [8,Question B]. He also mentioned that the counterexamples, if exist, seem to be rare.…”

Brauer characters, degrees and subgroups

“…G. Qian has proposed an intriguing conjecture in [7]: if 𝐺 is a finite group, g ∈ 𝐺, then there is some 𝜒 ∈ Irr(𝐺) such that 𝑜(g) divides |𝐺 ∶ ker(𝜒)|∕𝜒 (1). In his main result in [7], he proves this fact for solvable groups.…”

“…Qian has proposed an intriguing conjecture in [7]: if 𝐺 is a finite group, g ∈ 𝐺, then there is some 𝜒 ∈ Irr(𝐺) such that 𝑜(g) divides |𝐺 ∶ ker(𝜒)|∕𝜒 (1). In his main result in [7], he proves this fact for solvable groups. (A reduction theorem for this conjecture to a question on simple groups is proposed, among other results, in [4].…”