Character codegrees in finite groups

Abstract

For an irreducible character \chi of a finite group G, we define its codegree as \mathrm{c}\mathrm{o}\mathrm{d}\left(\chi \right)=\frac{|G:\ker \chi |}{\chi \left(1\right)}. In this paper, we introduce some known results and unsolved problems about character codegrees in finite groups.

Key words： Finite group; character; character codegree

Acknowledgements

I would like to express my gratitude to my colleagues for their valuable suggestions, especially to Professor Qinhai Zhang for his repeated encouragement. I am also grateful for the financial support provided by the National Natural Science Foundation of China (Grant No. 12171058).

References

1 Ahanjideh N. The Fitting subgroup, p-length, derived length and character table. Math Nachr 2021; 294(2): 214–223

2 Alizadeh F, Behravesh H, Ghaffarzadeh M, Ghasemi M, Hekmatara S. Groups with few codegrees of irreducible characters. Comm Algebra 2019; 47(3): 1147–1152

3 Alizadeh F, Ghasemi M, Ghaffarzadeh M. Finite groups whose codegrees are almost prime. Comm Algebra 2021; 49(2): 538–544

4 Bahramian R, Ahanjideh N. p-divisibility of co-degrees of irreducible characters. Bull Aust Math Soc 2021; 103(1): 78–82

5 Bahramian R, Ahanjideh N. p-parts of co-degrees of irreducible characters. Comptes Rendus Math Acad Sci Paris 2021; 359: 79–83

6 Bahri A, Akhlaghi Z, Khosravi B. An analogue of Huppert’s conjecture for character codegrees. Bull Aust Math Soc 2021; 104(2): 278–286

7 Chen X, Yang Y. Normal p-complements and monomial characters. Monatsh Math 2020; 193(4): 807–810

8 Chillag D, Herzog M. On character degree quotients. Arch Math (Basel) 1990; 55(1): 25–29

9 Chillag D, Mann A, Manz O. The codegrees of irreducible characters. Israel J Math 1991; 73(2): 207–223

10 Croome S, Lewis M L. Character codegrees of maximal class p-groups. Canad Math Bull 2020; 63(2): 328–334

11 Du N, Lewis M L. Codegrees and nilpotence class of p-groups. J Group Theory 2016; 19(4): 561–567

12 Gagola S M Jr, Lewis M L. A character theoretic condition characterizing nilpotent groups. Comm Algebra 1999; 27(3): 1053–1056

13 Gao Y, Liu Y. On codegrees and solvable groups. Bull Iranian Math Soc 2022; 48(4): 1357–1363

14 HuppertB. Character Theory of Finite Groups. De Gruyter Expositions in Mathematics, Vol 25. Berlin: Walter de Gruyter & Co, 1998

15 IsaacsI M. Character Theory of Finite Groups. NewYork: Academic Press, 1976

16 Isaacs I M. Element orders and character codegrees. Arch Math (Basel) 2011; 97(6): 499–501

17 Keller T M. Solvable groups with a small number of prime divisors in the element orders. J Algebra 1994; 170(2): 625–648

18 Keller T M. A linear bound for ρ(n). J Algebra 1995; 178(2): 643–652

19 Liang D, Qian G. Finite groups with coprime character degrees and codegrees. J Group Theory 2016; 19(5): 763–776

20 Liang D, Qian G, Shi W. Finite groups whose all irreducible character degrees are Hall-numbers. J Algebra 2007; 307(2): 695–703

21 Liu J, Wu K, Meng W. P-characters and the structure of finite solvable groups. J Group Theory 2021; 24(2): 285–291

22 Liu Y, Yang Y. Nonsolvable groups with four character codegrees. J Algebra Appl 2021; 20(11): Paper No 2150196 (5 pp)

23 Liu Y, Yang Y. Nonsolvable groups with five character codegrees. Comm Algebra 2021; 49(3): 1274–1279

24 Moretó A. Huppert’s conjecture for character codegrees. Math Nachr 2021; 294(11): 2232–2236

25 Qian G. Notes on character degrees quotients of finite groups. J Math 2002; 22(2): 217–220

26 Qian G. Character degrees quotients and structures of finite groups. Chinese Ann Math Ser A 2005; 26(3): 307–312

27 Qian G, Wang Y, Wei H. Co-degrees of irreducible characters in finite groups. J Algebra 2007; 312(2): 946–955

28 Qian G. A note on element orders and character codegrees. Arch Math (Basel) 2011; 97(2): 99–103

29 Qian G. A character theoretic criterion for p-closed group. Israel J Math 2012; 190: 401–412

30 Qian G. A note on p-parts of character degrees. Bull Lond Math Soc 2018; 50(4): 663–666

31 Qian G. Element orders and character codegrees. Bull Lond Math Soc 2021; 53(3): 820–824

32 Qian G, Yang Y. Nonsolvable groups with no prime dividing three character degrees. J Algebra 2015; 436: 145–160

33 Qian G, Zhou Y. On p-parts of character codegrees. J Group Theory 2021; 24(5): 1005–1018

34 Ren Y. On character degrees quotients of finite groups. Acta Math Sinica Chin Ser 1995; 38(2): 164–170

35 Sayanjali Z, Akhlaghi Z, Khosravi B. On the codegrees of finite groups. Comm Algebra 2020; 48(3): 1327–1332

36 Tong-Viet H P. Brauer characters and normal Sylow p-subgroups. J Algebra 2018; 503: 265–276

37 Xiong H. Finite groups whose character graphs associated with codegrees have no triangles. Algebra Colloq 2016; 23(1): 15–22

38 Yang D, Gao C, Lv H. Finite groups with special codegrees. Arch Math (Basel) 2020; 115(2): 121–130

39 Yang Y. On analogues of Huppert’s conjecture. Bull Aust Math Soc 2021; 104(2): 272–277

40 Yang Y, Qian G. The analog of Huppert’s conjecture on character codegrees. J Algebra 2017; 478: 215–219

41 Yang Y, Qian G. On p-parts of character degrees and conjugacy class sizes of finite groups. Adv Math 2018; 328: 356–366

42 Zhang J. Arithmetical conditions on element orders and group structure. Proc Amer Math Soc 1995; 123(1): 39–44