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Weights for 𝜋-partial characters of 𝜋-separable groups

  • Xuewu Chang EMAIL logo and Ping Jin
Published/Copyright: November 7, 2024

Abstract

The aim of this paper is to confirm a conjecture of Isaacs and Navarro from 1995, which asserts that, for any π -subgroup 𝑄 of a 𝜋-separable group 𝐺, the number of π -weights of 𝐺 with 𝑄 as the first component is larger than or equal to the number of irreducible 𝜋-partial characters of 𝐺 with 𝑄 as their vertex. We also give a sufficient condition to guarantee that these two numbers are equal, and thereby strengthen their main theorem on the 𝜋-version of the Alperin weight conjecture.

Award Identifier / Grant number: 12171289

Funding statement: This work was supported by the NSF of China (12171289) and by Fundamental Research Programs of Shanxi Province (20210302123429 and 20210302124077).

Acknowledgements

The authors are grateful to the referee for valuable comments and suggestions that greatly helped us to improve the presentation of this paper.

  1. Communicated by: Hung Tong-Viet

References

[1] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976. Search in Google Scholar

[2] I. M. Isaacs, Finite Group Theory, American Mathematical Society, Providence, 2008. 10.1090/gsm/092Search in Google Scholar

[3] I. M. Isaacs, Characters of Solvable Groups, American Mathematical Society, Providence, 2018. 10.1090/gsm/189Search in Google Scholar

[4] I. M. Isaacs and G. Navarro, Weights and vertices for characters of 𝜋-separable groups, J. Algebra 177 (1995), 339–366. 10.1006/jabr.1995.1301Search in Google Scholar

[5] M. Lewis, Characters, coprime actions, and operator groups, Arch. Math. 69 (1997), 455–460. 10.1007/s000130050145Search in Google Scholar

[6] G. Navarro, Vertices for characters of 𝑝-solvable groups, Trans. Amer. Math. Soc. 365 (2002), 2759–2773. 10.1090/S0002-9947-02-02974-4Search in Google Scholar

[7] G. Navarro and B. Sambale, Weights and nilpotent subgroups, Int. Math. Res. Not. IMRN 2021 (2021), 2526–2538. 10.1093/imrn/rnz195Search in Google Scholar

[8] A. Turull, Above the Glauberman correspondence, Adv. Math. 217 (2008), 2170–2205. 10.1016/j.aim.2007.10.001Search in Google Scholar

[9] T. Wolf, Variations on McKay’s character degree conjecture, J. Algebra 135 (1990), 123–138. 10.1016/0021-8693(90)90153-FSearch in Google Scholar

Received: 2024-05-23
Revised: 2024-09-30
Published Online: 2024-11-07
Published in Print: 2025-05-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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