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Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

Postal Subscription Code 80-964

2018 Impact Factor: 0.565

Front. Math. China    2023, Vol. 18 Issue (1) : 15-32    https://doi.org/10.3868/S140-DDD-023-006-X
SURVEY ARTICLE
Character codegrees in finite groups
Guohua QIAN()
School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China
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Abstract

For an irreducible character χ of a finite group G, we define its codegree as cod(χ)=|G:kerχ|χ(1). In this paper, we introduce some known results and unsolved problems about character codegrees in finite groups.

Keywords Finite group      character      character codegree     
Online First Date: 22 May 2023    Issue Date: 31 May 2023
 Cite this article:   
Guohua QIAN. Character codegrees in finite groups[J]. Front. Math. China, 2023, 18(1): 15-32.
 URL:  
https://academic.hep.com.cn/fmc/EN/10.3868/S140-DDD-023-006-X
https://academic.hep.com.cn/fmc/EN/Y2023/V18/I1/15
1 N Ahanjideh. The Fitting subgroup, p-length, derived length and character table. Math Nachr 2021; 294(2): 214–223
2 F Alizadeh, H Behravesh, M Ghaffarzadeh, M Ghasemi, S Hekmatara. Groups with few codegrees of irreducible characters. Comm Algebra 2019; 47(3): 1147–1152
3 F Alizadeh, M Ghasemi, M Ghaffarzadeh. Finite groups whose codegrees are almost prime. Comm Algebra 2021; 49(2): 538–544
4 R Bahramian, N Ahanjideh. p-divisibility of co-degrees of irreducible characters. Bull Aust Math Soc 2021; 103(1): 78–82
5 R Bahramian, N Ahanjideh. p-parts of co-degrees of irreducible characters. Comptes Rendus Math Acad Sci Paris 2021; 359: 79–83
6 A Bahri, Z Akhlaghi, B Khosravi. An analogue of Huppert’s conjecture for character codegrees. Bull Aust Math Soc 2021; 104(2): 278–286
7 X Chen, Y Yang. Normal p-complements and monomial characters. Monatsh Math 2020; 193(4): 807–810
8 D Chillag, M Herzog. On character degree quotients. Arch Math (Basel) 1990; 55(1): 25–29
9 D Chillag, A Mann, O Manz. The codegrees of irreducible characters. Israel J Math 1991; 73(2): 207–223
10 S Croome, M L Lewis. Character codegrees of maximal class p-groups. Canad Math Bull 2020; 63(2): 328–334
11 N Du, M L Lewis. Codegrees and nilpotence class of p-groups. J Group Theory 2016; 19(4): 561–567
12 S M Jr Gagola, M L Lewis. A character theoretic condition characterizing nilpotent groups. Comm Algebra 1999; 27(3): 1053–1056
13 Y Gao, Y Liu. On codegrees and solvable groups. Bull Iranian Math Soc 2022; 48(4): 1357–1363
14 B Huppert. Character Theory of Finite Groups. De Gruyter Expositions in Mathematics, Vol 25. Berlin: Walter de Gruyter & Co, 1998
15 I M Isaacs. Character Theory of Finite Groups. NewYork: Academic Press, 1976
16 I M Isaacs. Element orders and character codegrees. Arch Math (Basel) 2011; 97(6): 499–501
17 T M Keller. Solvable groups with a small number of prime divisors in the element orders. J Algebra 1994; 170(2): 625–648
18 T M Keller. A linear bound for ρ(n). J Algebra 1995; 178(2): 643–652
19 D Liang, G Qian. Finite groups with coprime character degrees and codegrees. J Group Theory 2016; 19(5): 763–776
20 D Liang, G Qian, W Shi. Finite groups whose all irreducible character degrees are Hall-numbers. J Algebra 2007; 307(2): 695–703
21 J Liu, K Wu, W Meng. P-characters and the structure of finite solvable groups. J Group Theory 2021; 24(2): 285–291
22 Y Liu, Y Yang. Nonsolvable groups with four character codegrees. J Algebra Appl 2021; 20(11): Paper No 2150196 (5 pp)
23 Y Liu, Y Yang. Nonsolvable groups with five character codegrees. Comm Algebra 2021; 49(3): 1274–1279
24 A Moretó. Huppert’s conjecture for character codegrees. Math Nachr 2021; 294(11): 2232–2236
25 G Qian. Notes on character degrees quotients of finite groups. J Math 2002; 22(2): 217–220
26 G Qian. Character degrees quotients and structures of finite groups. Chinese Ann Math Ser A 2005; 26(3): 307–312
27 G Qian, Y Wang, H Wei. Co-degrees of irreducible characters in finite groups. J Algebra 2007; 312(2): 946–955
28 G Qian. A note on element orders and character codegrees. Arch Math (Basel) 2011; 97(2): 99–103
29 G Qian. A character theoretic criterion for p-closed group. Israel J Math 2012; 190: 401–412
30 G Qian. A note on p-parts of character degrees. Bull Lond Math Soc 2018; 50(4): 663–666
31 G Qian. Element orders and character codegrees. Bull Lond Math Soc 2021; 53(3): 820–824
32 G Qian, Y Yang. Nonsolvable groups with no prime dividing three character degrees. J Algebra 2015; 436: 145–160
33 G Qian, Y Zhou. On p-parts of character codegrees. J Group Theory 2021; 24(5): 1005–1018
34 Y Ren. On character degrees quotients of finite groups. Acta Math Sinica Chin Ser 1995; 38(2): 164–170
35 Z Sayanjali, Z Akhlaghi, B Khosravi. On the codegrees of finite groups. Comm Algebra 2020; 48(3): 1327–1332
36 H P Tong-Viet. Brauer characters and normal Sylow p-subgroups. J Algebra 2018; 503: 265–276
37 H Xiong. Finite groups whose character graphs associated with codegrees have no triangles. Algebra Colloq 2016; 23(1): 15–22
38 D Yang, C Gao, H Lv. Finite groups with special codegrees. Arch Math (Basel) 2020; 115(2): 121–130
39 Y Yang. On analogues of Huppert’s conjecture. Bull Aust Math Soc 2021; 104(2): 272–277
40 Y Yang, G Qian. The analog of Huppert’s conjecture on character codegrees. J Algebra 2017; 478: 215–219
41 Y Yang, G Qian. On p-parts of character degrees and conjugacy class sizes of finite groups. Adv Math 2018; 328: 356–366
42 J Zhang. Arithmetical conditions on element orders and group structure. Proc Amer Math Soc 1995; 123(1): 39–44
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