Research ArticleNo Access

On the average codegree of a finite group

Zhongbi Wang

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, P. R. China

E-mail Address: zbwango1@163.com

Search for more papers by this author

Guohua Qian

Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, P. R. China

E-mail Address: ghqian2000@163.com

Search for more papers by this author

Heng Lv

School of Mathematics and Statistics, Southwest University, Beibei, Chongqing 400715, P. R. China

E-mail Address: lvh529@swu.edu.cn

Search for more papers by this author

, and

Guiyun Chen

https://orcid.org/0000-0002-3947-5295

School of Mathematics and Statistics, Southwest University, Beibei, Chongqing 400715, P. R. China

E-mail Address: gychen1963@163.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219498824501020Cited by:1 (Source: Crossref)

Abstract

Let G be a finite group, we define the average codegree of the irreducible characters of G as $acod (G) = \frac{1}{| Irr (G) |} \sum_{χ \in Irr (G)} cod (χ)$ , where $cod (χ) = \frac{| G : ker χ |}{χ (1)}$ . We prove that if $G$ is non-solvable, then $acod (G) \geq 68 / 5$ , and the equality holds if and only if $G ≅ A_{5}$ . Also, we show that if $G$ is non-supersolvable, then $acod (G) \geq 11 / 4$ , and the equality holds if and only if $G ≅ A_{4}$ . In addition, we obtain that if $p$ is the smallest prime divisor of $| G |$ , then $acod (G) < p$ if and only if $G$ is an elementary abelian $p$ -group.

Communicated by M. L. Lewis

Keywords:

AMSC: 20C15, 20D10

References

1. F. Alizadeh, H. Behravesh, M. Ghaffarzadeh, M. Ghasemi and S. Hekmatara , Groups with few codegrees of irreducible characters, Commun. Algebra 47(3) (2019) 1147–1152. Crossref, Web of Science, Google Scholar
2. R. Bahramian and N. Ahanjideh , $p$ -divisibility of co-degrees of irreducible characters, Bull. Austral. Math. Soc. 103(1) (2021) 78–82. Crossref, Web of Science, Google Scholar
3. M. Bianchi, D. Chillag, M. L. Lewis and E. Pacifici , Character degree graphs that are complete graphs, Proc. Amer. Math. Soc. 135(3) (2007) 671–676. Crossref, Web of Science, Google Scholar
4. J. H. Conway, R. T. Curtis and S. P. Norton , Atlas of Finite Groups (Clarendon Press, Oxford, UK, 1985). Google Scholar
5. S. Croome and M. L. Lewis , Character codegrees of maximal class $p$ -groups, Canad. Math. Bull. 63(2) (2020) 328–334. Crossref, Web of Science, Google Scholar
6. N. Du and M. L. Lewis , Codegrees and nilpotence class of $p$ -groups, J. Group Theory 19(4) (2016) 561–567. Crossref, Web of Science, Google Scholar
7. N. N. Hung and P. H. Tiep , The average character degree and an improvement of the Itǒ–Michler theorem, J. Algebra 550 (2020) 86–107. Crossref, Web of Science, Google Scholar
8. I. M. Isaacs , Character Theory of Finite Groups (Dover, 1994). Google Scholar
9. I. M. Isaacs, M. Loukaki and A. Moretó , The average degree of an irreducible character of a finite group, Israel J. Math. 197(1) (2013) 55–67. Crossref, Web of Science, Google Scholar
10. T. M. Keller and Y. Yang , Abelian quotients and orbit sizes of solvable linear groups, Israel J. Math. 211(1) (2016) 23–44. Crossref, Web of Science, Google Scholar
11. A. Moretó and H. N. Nguyen , On the average character degree of finite groups, Bull. London Math. Soc. 46(3) (2014) 454–462. Crossref, Web of Science, Google Scholar
12. G. H. Qian , A character theoretic criterion for a $p$ -closed group, Israel J. Math. 190 (2012) 401–412. Crossref, Web of Science, Google Scholar
13. G. H. Qian , On the average character degree and the average class size in finite groups, J. Algebra 423 (2015) 1191–1212. Crossref, Web of Science, Google Scholar
14. G. H. Qian, Y. M. Wang and H. Q. Wei , Co-degrees of irreducible characters in finite groups, J. Algebra 312(2) (2007) 946–955. Crossref, Web of Science, Google Scholar