Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter January 23, 2022

A dual version of Huppert's ρ-σ conjecture for character codegrees

Alexander Moretó ORCID logo
From the journal Forum Mathematicum

Abstract

We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if k is a positive integer such that for any prime p the number of character codegrees of a finite group G that are divisible by p is at most k, then the number of prime divisors of |G| is bounded in terms of k. We prove this conjecture for solvable groups.

MSC 2010: 20C15

Communicated by Freydoon Shahidi


Funding source: Ministerio de Ciencia e Innovación

Award Identifier / Grant number: PID2019-103854GB-I00

Funding source: Generalitat Valenciana

Award Identifier / Grant number: AICO/2020/298

Funding statement: Research supported by Ministerio de Ciencia e Innovación (Grant PID2019-103854GB-I00 funded by MCIN/AEI/10.13039/501100011033) and Generalitat Valenciana AICO/2020/298.

Acknowledgements

I thank the referee for his/her careful reading of the paper and helpful comments.

References

[1] N. Ahanjideh, The fitting subgroup, p-length, derived length and character table, Math. Nachr. 294 (2021), no. 2, 214–223. 10.1002/mana.202000057Search in Google Scholar

[2] Z. Akhlaghi, S. Dolfi and E. Pacifici, On Huppert’s rho-sigma conjecture, J. Algebra 586 (2021), 537–560. 10.1016/j.jalgebra.2021.06.038Search in Google Scholar

[3] F. Alizadeh, H. Behravesh, M. Ghaffarzadeh, M. Ghasemi and S. Hekmatara, Groups with few codegrees of irreducible characters, Comm. Algebra 47 (2019), no. 3, 1147–1152. 10.1080/00927872.2018.1501572Search in Google Scholar

[4] D. Benjamin, Coprimeness among irreducible character degrees of finite solvable groups, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2831–2837. 10.1090/S0002-9939-97-04269-XSearch in Google Scholar

[5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Clarendon Press, Oxford, 1985. Search in Google Scholar

[6] N. Du and M. L. Lewis, Codegrees and nilpotence class of p-groups, J. Group Theory 19 (2016), no. 4, 561–567. 10.1515/jgth-2015-0039Search in Google Scholar

[7] I. M. Isaacs, Character Theory of Finite Groups, AMS Chelsea, Providence, 2006. 10.1090/chel/359Search in Google Scholar

[8] T. M. Keller, Orbit sizes and character degrees. III, J. Reine Angew. Math. 545 (2002), 1–17. 10.1515/crll.2002.035Search in Google Scholar

[9] G. Malle and A. Moretó, A dual version of Huppert’s ρ-σ conjecture, Int. Math. Res. Not. IMRN 2007 (2007), no. 22, Article ID rnm 104. Search in Google Scholar

[10] O. Manz and T. R. Wolf, Representations of Solvable Groups, London Math. Soc. Lecture Note Ser. 185, Cambridge University, Cambridge, 1993. 10.1017/CBO9780511525971Search in Google Scholar

[11] S. Mattarei, On character tables of wreath products, J. Algebra 175 (1995), no. 1, 157–178. 10.1006/jabr.1995.1180Search in Google Scholar

[12] A. Moretó, Character degrees, character codegrees and nilpotence class of p-groups, Comm. Algebra (2021), 10.1080/00927872.2021.1970758. 10.1080/00927872.2021.1970758Search in Google Scholar

[13] A. Moretó, Huppert’s conjecture for character codegrees, Math. Nachr. 294 (2021), no. 11, 2232–2236. 10.1002/mana.202000568Search in Google Scholar

[14] G. Qian, Y. Wang and H. Wei, Co-degrees of irreducible characters in finite groups, J. Algebra 312 (2007), no. 2, 946–955. 10.1016/j.jalgebra.2006.11.001Search in Google Scholar

[15] J. S. Williams, Prime graph components of finite groups, J. Algebra 69 (1981), no. 2, 487–513. 10.1090/pspum/037/604579Search in Google Scholar

[16] Y. Yang and G. Qian, The analog of Huppert’s conjecture on character codegrees, J. Algebra 478 (2017), 215–219. 10.1016/j.jalgebra.2016.12.017Search in Google Scholar

Received: 2021-10-05
Revised: 2021-12-09
Published Online: 2022-01-23
Published in Print: 2022-03-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston