A note on element orders and character codegrees

Qian, Guohua

doi:10.1007/s00013-011-0278-6

Cited by 23 publications

(21 citation statements)

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“…On the other hand, by [8] and [6] (or [12]), As the proof above illustrates, this "gluing" process is an obstacle to obtaining best possible bounds in any variation of the 𝜌-𝜎 conjecture for arbitrary groups. It would be interesting to find some new idea that avoids this argument.…”

Section: Proofsmentioning

confidence: 99%

Huppert's conjecture for character codegrees

Moretó

2021

Mathematische Nachrichten

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Huppert's ρ‐σ conjecture is one of the main open problems on character degrees of finite groups. A number of ρ‐σ problems have been studied. For instance, T. Keller and J. Zhang considered the ρ‐σ problem for element orders in the 1990s. Recently, a lot of research is being done on character codegrees. Y. Yang and G. Qian studied the ρ‐σ problem for character codegrees in 2017. In this note, we obtain a sharp bound for groups with trivial solvable radical. As a consequence, we improve the general bound of Yang and Qian. For solvable groups, we notice that the ρ‐σ problem for character codegrees is very closely related to the ρ‐σ problem for element orders. In particular, we give a partial negative answer to a speculation by Yang and Qian on the exact bound for the ρ‐σ problem for character codegrees.

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Section: Proofsmentioning

confidence: 99%

Huppert's conjecture for character codegrees

Moretó

2021

Mathematische Nachrichten

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“…/ D 1 if and only if D 1 G . Given [5] and [9], it is not surprising that the exponent of the derived factor group will influence the codegrees occurring in cod.G/.…”

Section: Basicsmentioning

confidence: 99%

“…In [9], Qian showed that if G is a solvable group and g 2 G, then there exists 2 Irr.G/ so that every prime divisor of o.g/ divides cod. /.…”

Section: Introductionmentioning

confidence: 99%

Codegrees and nilpotence class of p-groups

Lewis

2015

Journal of Group Theory

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“…In his recent paper [2] in this journal, Guohua Qian proved the following theorem under the additional hypothesis that the group G is solvable. The purpose of this note is to give an easy proof with no solvability assumption.…”

Section: Mathematics Subject Classification (2010) 20c15mentioning

confidence: 99%

Element orders and character codegrees

Isaacs

2011

Arch. Math.

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Let g ∈ G, where G is an arbitrary finite group. Then there exists χ ∈ Irr(G) such that ker(χ) ∩ g = 1 and every prime divisor of the order o(g) divides the codegree of χ. This improves a recent result of Qian, in which G was assumed to be solvable. Mathematics Subject Classification (2010). 20C15.In his recent paper [2] in this journal, Guohua Qian proved the following theorem under the additional hypothesis that the group G is solvable. The purpose of this note is to give an easy proof with no solvability assumption. To state the result, we recall Qian's definition: the codegree of a character χ ∈ Irr(G) is the integer |G : ker(χ)|/χ(1).where G is a finite group. Then there exists a character χ ∈ Irr(G) with ker(χ) ∩ g = 1, and such that every prime divisor of the order o(g) divides the codegree of χ.The key to our proof is the following.Lemma. Let n be a square-free positive integer. Then the set X of primitive n th roots of unity in the complex field C is linearly independent over the rational field Q.Proof. Suppose that a δ δ = 0, where a δ ∈ Q and the sum runs over δ ∈ X. We show by induction on the number k of prime divisors of n that a δ = 0 for all δ ∈ X. If k = 0 then n = 1 and X = {1}, and the result clearly holds in this case. We can thus assume that k > 0, and we let p be a prime divisor of n. Since n is square-free, each element δ ∈ X is uniquely of the form δ = μν, where μ is a primitive p th root of unity and ν is a primitive m th root of unity, where m = n/p. Let Y and Z, respectively, be the sets of all primitive p th and m th roots of unity.

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A note on element orders and character codegrees

Abstract: The result of this note is as follows. If a finite solvable group has an element of order m, then the group has an irreducible character whose codegree contains all prime divisors of m.Mathematics Subject Classification (2010). Primary 20C20, 20C15.

Cited by 23 publications

References 5 publications

Huppert's conjecture for character codegrees

Huppert's conjecture for character codegrees

Codegrees and nilpotence class of p-groups

Element orders and character codegrees

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