no. G6
Algèbre
On the number of prime divisors of character degrees and conjugacy classes of a finite group
Yang, Yong1, 2 
1 Three Gorges Mathematical Research Center, College of Science, China Three Gorges University, Yichang, Hubei 443002, China
2 Department of Mathematics, Texas State University, San Marcos, TX 78666, USA
Comptes Rendus. Mathématique, Tome 360 (2022) no. G6, pp. 583-588.

A result of Gluck is that any finite group G has an abelian subgroup A such that |G:A| is bounded by a polynomial function of the largest irreducible character degree of G. Moretó presented a variation of this result that looks at the number of prime factors of the irreducible character degrees and obtained an almost quadratic bound. The author improved the result of Moretó to almost linear. In this note, we further improve the bound, and also study the related problem on conjugacy class sizes.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.301
Classification : 20C15, 20C20
Yang, Yong 1, 2

1 Three Gorges Mathematical Research Center, College of Science, China Three Gorges University, Yichang, Hubei 443002, China
2 Department of Mathematics, Texas State University, San Marcos, TX 78666, USA 
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Yang, Yong. On the number of prime divisors of character degrees and conjugacy classes of a finite group. Comptes Rendus. Mathématique, Tome 360 (2022) no. G6, pp. 583-588. doi : 10.5802/crmath.301. http://www.numdam.org/articles/10.5802/crmath.301/

[1] Burkett, Shawn T.; Nguyen, Hung Ngoc Conjugacy classes of small sizes in the linear and unitary groups, J. Group Theory, Volume 16 (2013) no. 6, pp. 851-874 | DOI | MR | Zbl

[2] Conway, John H.; Curtis, Robert T.; Norton, Simon P.; Parker, Richard A.; Wilson, Robert A. Atlas of Finite Groups. Maximal subgroups and ordinary characters for simple groups, Clarendon Press, 1985 (with comput. assist. from J. G. Thackray) | Zbl

[3] Dornhoff, Larry Group Representation Theory (in 2 parts). Part A: Ordinary representation theory, Pure and Applied Mathematics, 7, Marcel Dekker, 1971 | MR | Zbl

[4] Gagola, Stephen M. jun. A character theoretic condition for F(G)>1, Commun. Algebra, Volume 33 (2005) no. 5, pp. 1369-1382 | DOI | MR | Zbl

[5] Gasarch, William; Washington, Larry ∑ p≤n 1/p=ln(lnn)+O(1) : An Exposition (2015) (https://arxiv.org/abs/1511.01823)

[6] Gluck, David The largest irreducible character degree of a finite group, Can. J. Math., Volume 37 (1985), pp. 442-451 | DOI | MR | Zbl

[7] Lubeck, Frank Data for Finite Groups of Lie Type and Related Algebraic Groups (http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/index.html)

[8] Moretó, Alexander A variation on theorems of Jordan and Gluck, J. Algebra, Volume 301 (2006) no. 1, pp. 274-279 | DOI | MR | Zbl

[9] Nguyen, Hung Ngoc Low-dimensional complex characters of the symplectic and orthogonal groups, Commun. Algebra, Volume 38 (2010) no. 3, pp. 1157-1197 | DOI | MR | Zbl

[10] Tiep, Pham Huu; Zalesskiĭ, Alexandre E. Unipotent elements of finite groups of Lie type and realization fields of their complex representations, J. Algebra, Volume 271 (2004) no. 1, pp. 327- 390 | DOI | MR | Zbl

[11] Yang, Yong Orbits of the actions of finite solvable groups, J. Algebra, Volume 321 (2009) no. 7, pp. 2012-2021 | DOI | MR | Zbl

[12] Yang, Yong A variation on a theorem of Gluck, Monatsh. Math., Volume 185 (2018) no. 1, pp. 159-162 | DOI | MR | Zbl

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