Character degree sums of finite groups

Maróti, Attila; Nguyen, Hung Ngoc

doi:10.1515/forum-2013-0066

Cited by 12 publications

(2 citation statements)

References 14 publications

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“…Theorem 1 also significantly improves a result of J. P. Cossey and the second author [3, Theorem A] that if S a non-abelian composition factor of G different from the simple linear groups PSL 2 (q), then the number of times that S occurs as a composition factor of G is bounded in terms of rat(G). We refer the reader to [3,7,8,10] for more discussion on the influence of the character degree ratio and character degrees in general on the structure of finite groups.…”

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confidence: 99%

The character degree ratio and composition factors of a finite group

Lewis

Nguyen

2015

Monatsh Math

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Section: Introductionmentioning

confidence: 99%

The character degree ratio and composition factors of a finite group

Lewis

Nguyen

2015

Monatsh Math

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“…The degree t (G) has also been used in a number of recent papers to determine other properties of G such as whether G is solvable, supersolvable, nilpotent, etc. (See [9], [13], [14], [19].) In [1], they study the difference t (G) − t (H ) where H is a proper subgroup of G. In [12], they compare t (G) and t (G/N) when N is a nonsolvable group.…”

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On the Existence of Johnson Polynomials for Nilpotent Groups

Lewis

Prajapati

2014

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Let G be a finite group. We say that G has a Johnson polynomial if there exists a polynomial f (x) ∈ Z[x] and a character χ ∈ Irr(G) so that f (χ) equals the total character for G. In this paper, we show that if G has nilpotence class 2, then G has a Johnson polynomial if and only if G is an extra-special 2-group. Generalizing this, we say that G has a generalized Johnson polynomial if f (x) ∈ Q[x]. We show that if G has nilpotence class 2, then G has a generalized Johnson polynomial if and only if Z(G) is cyclic. Also, if G is nilpotent and | cd(G)| = 2, then G has a generalized Johnson polynomial if and only if G has nilpotence class 2 and Z(G) is cyclic.

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A Note on the Codegree of Finite Groups

Lewis,

Yan

2024

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Let $$\chi $$ χ be an irreducible character of a group G, and $$S_c(G)=\sum _{\chi \in \textrm{Irr}(G)}\textrm{cod}(\chi )$$ S c ( G ) = ∑ χ ∈ Irr ( G ) cod ( χ ) be the sum of the codegrees of the irreducible characters of G. Write $$\textrm{fcod} (G)=\frac{S_c(G)}{|G|}.$$ fcod ( G ) = S c ( G ) | G | . We aim to explore the structure of finite groups in terms of $$\textrm{fcod} (G).$$ fcod ( G ) . On the other hand, we determine the lower bound of $$S_c(G)$$ S c ( G ) for nonsolvable groups and prove that if G is nonsolvable, then $$S_c(G)\geqslant S_c(A_5)=68,$$ S c ( G ) ⩾ S c ( A 5 ) = 68 , with equality if and only if $$G\cong A_5.$$ G ≅ A 5 . Additionally, we show that there is a solvable group so that it has the codegree sum as $$A_5.$$ A 5 .

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Character degree sums of finite groups

Abstract: Abstract. We present some results on character degree sums in connection with several important characteristics of finite groups such as p-solvability, solvability, supersolvability, and nilpotency. Some of them strengthen known results in the literature.

Cited by 12 publications

References 14 publications

The character degree ratio and composition factors of a finite group

The character degree ratio and composition factors of a finite group

On the Existence of Johnson Polynomials for Nilpotent Groups

A Note on the Codegree of Finite Groups

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