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5-Regular prime graphs of finite nonsolvable groups

  • Qinghong Guo , Weijun Liu EMAIL logo and Lu Lu
Published/Copyright: June 8, 2023

Abstract

The prime graph Ξ” ⁒ ( G ) of a finite group 𝐺 is a graph whose vertex set is the set of prime factors of the degrees of all irreducible complex characters of 𝐺, and two distinct primes 𝑝 and π‘ž are joined by an edge if the product p ⁒ q divides some character degree of 𝐺. In 2014, Tong-Viet [H. P. Tong-Viet, Finite groups whose prime graphs are regular, J. Algebra 397 (2014), 18–31] proposed the following conjecture. Let 𝐺 be a group and let k β‰₯ 5 be odd. If the prime graph Ξ” ⁒ ( G ) is π‘˜-regular, then Ξ” ⁒ ( G ) is a complete graph of order k + 1 . In this paper, we show that if the prime graph Ξ” ⁒ ( G ) of a finite nonsolvable group 𝐺 is 5-regular, then Ξ” ⁒ ( G ) is isomorphic to the complete graph K 6 or possibly the graph depicted in the first figure below. Moreover, if 𝐺 is an almost simple group, then Ξ” ⁒ ( G ) is isomorphic to the complete graph K 6 .

Award Identifier / Grant number: 12071484

Award Identifier / Grant number: 12271524

Award Identifier / Grant number: 2022JJ30674

Funding statement: This research was supported by NSFC (Nos. 12071484 and 12271524), NSF of Hunan (No. 2022JJ30674) and the Foundation of Guangdong University of Science and Technology.

Acknowledgements

We would like to thank Prof. Lihua Feng for his careful reading of this paper and for the helpful suggestions in its writing. We would also like to thank the editor and reviewers for their valuable suggestions and useful comments.

  1. Communicated by: Hung Tong-Viet

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Received: 2023-03-08
Revised: 2023-04-05
Published Online: 2023-06-08
Published in Print: 2023-11-01

Β© 2023 Walter de Gruyter GmbH, Berlin/Boston

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