Journal of Guangxi Normal University(Natural Science Edition) ›› 2022, Vol. 40 ›› Issue (5): 160-167.doi: 10.16088/j.issn.1001-6600.2022020703

Previous Articles     Next Articles

Codegrees of Irreducible Characters in Finite Groups

LU Jiakuan*, WANG Yu, ZHANG Boru, PANG Linna   

  1. School of Mathematics and Statistics, Guangxi Normal University, Guilin Guangxi 541006, China
  • Received:2022-02-07 Revised:2022-03-22 Online:2022-09-25 Published:2022-10-18

PDF (PC)

298

Abstract: Let G be a finite group, and χ∈Irr(G). The number cod(χ)=|G∶ker χ|χ(1) is called the codegree of the character χ. This paper makes a review on the results concerning codegree of irreducible character in finite groups in particular, lists some problems which are still open.

Key words: irreducible character, codegree, solvable groups, p-length, Huppert’s conjecture

CLC Number: 

  • O152.1
[1]ISAACS I M. Character theory of finite groups[M]. Providence,RI: AMS Chelsea Publishing, 2006.
[2]BERKOVICH Y G, ZHMUD E M. Characters of finite groups. Part 1.[M]. Providence: American Mathematical Society, 1998.
[3]BERKOVICH Y G, ZHMUD E M. Characters of finite groups. Part 2.[M]. Providence: American Mathematical Society, 1999.
[4]ISAACS I M. Characters of solvable groups[M]. Providence: American Mathematical Society, 2018.
[5]QIAN G H, WANG Y M, WEI H Q. Co-degrees of irreducible characters in finite groups[J]. Journal of Algebra, 2007, 312(2): 946-955.
[6]CHILLAG D, HERZOG M. On character degrees quotients[J]. Archiv der Mathematik, 1990, 55(1): 25-29.
[7]CHILLAG D, MANN A, MANZ O. The co-degrees of irreducible characters[J]. Israel Journal of Mathematics,1991, 73(2): 207-223.
[8]任永才. 关于有限群的特征标次数的商[J]. 数学学报, 1995, 38(2): 164-170.
[9]钱国华. 特征标的算术条件与有限群结构[D]. 武汉:武汉大学, 2002.
[10]钱国华. 有限群的特征标与共轭类[D]. 广州:中山大学, 2007.
[11]钱国华. 有限群特征标次数商的几点注记[J]. 数学杂志, 2002, 22(2): 217-220.
[12]钱国华. 特征标次数的商与有限群结构[J]. 数学年刊A辑, 2005,26(3): 307-312.
[13]GAGOLA S M, Jr,LEWIS M L. A character theoretic condition characterizing nilpotent groups[J]. Communications in Algebra, 1999, 27(3): 1053-1056.
[14]QIAN G H. A character theoretic criterion for a p-closed group[J]. Israel Journal of Mathematics, 2012, 190(1): 401-412.
[15]LIANG D F, QIAN G H, SHI W J. Finite groups whose all irreducible character degrees are Hall-numbers[J]. Journal of Algebra, 2007, 307(2): 695-703.
[16]LIANG D F, QIAN G H. Finite groups with coprime character degrees and codegrees[J]. Journal of Group Theory,2016, 19(5): 763-776.
[17]BERKOVICH Y. Degrees, kernels and quasikernels of monolithic characters[J]. Proceedings of the American Mathematical Society,2000, 128(11): 3211-3220.
[18]YANG D F, GAO C, LV H. Finite groups with special codegrees[J]. Archiv der Mathematik, 2020, 115(2): 121-130.
[19]GAO T L, LIU Y. On codegrees and solvable groups[J/OL]. Bulletin of the Iranian Mathematical Society:1-7[2021-06-07]. https://doi.org/10.1007/s41980-021-00587-7.
[20]THOMPSON J G. Normal p-complements and irreducible characters[J]. Journal of Algebra, 1970, 14(2): 129-134.
[21]MICHLER G O. Brauer’s conjectures and the classification of finite simple groups[C]. Representation Theory II Groups and Orders. Berlin: Springer, Heidelberg, 1986:129-142.
[22]CHEN X Y, YANG Y. Normal p-complements and monomial characters[J]. Monatshefte für Mathematik, 2020, 193(4): 807-810.
[23]CHEN X Y, YANG Y. Notes on normal p-complements and monomial characters[J/OL]. Quaestiones Mathematicae: 1-4[2021-10-27]. https://doi.org/10.2989/16073606.2021.1977409.
[24]QIAN G H, YANG Y. Permutation characters in finite solvable groups[J]. Communications in Algebra, 2018, 46(1) : 167-175.
[25]LU J K, WU K X, MENG W. p-characters and the structure of finite solvable groups[J]. Journal of Group Theory, 2021, 24(2): 285-291.
[26]BAHRAMIAN R, AHANJIDEH N. p-divisibility of co-degrees of irreducible characters[J]. Bulletin of the Australian Mathematical Society, 2021, 103(1): 78-82.
[27]LEWIS M L, NAVARRO G, TIEP P H, et al. p-parts of character degrees[J]. Bulletin of the London Mathematical Society, 2015, 92(2): 483-497.
[28]LEWIS M L, NAVARRO G, WOLF T R. p-parts of character degrees and the index of the fitting subgroup[J]. Journal of Algebra, 2014, 411: 182-190.
[29]QIAN G H. A note on p-parts of character degrees[J]. Bulletin of the London Mathematical Society, 2018, 50(4): 663-666.
[30]BAHRAMIAN R, AHANJIDEH N. p-parts of co-degrees of irreducible characters[J]. Comptes Rendus. Mathématique, 2021, 359(1): 79-83.
[31]QIAN G H, ZHOU Y Y. On p-parts of character codegrees[J]. Journal of Group Theory, 2021, 24(5): 1005-1018.
[32]NORITZSCH T. Groups having three complex irreducible character degrees[J]. Journal of Algebra, 1995, 175(3): 767-798.
[33]MALLEG, MORETÓ A. Nonsolvable groups with few character degrees[J]. Journal of Algebra, 2005, 294(1): 117-126.
[34]DU N, LEWIS M L. Codegrees and nilpotence class of p-groups[J]. Journal of Group Theory, 2016, 19(4): 561-567.
[35]ALIZADEH F, BEHRAVESH H, GHAFFARZADEH M, et al. Groups with few codegrees of irreducible characters[J]. Communications in Algebra, 2019, 47(3): 1147-1152.
[36]CROOMES, LEWIS M L. p-groups with exactly four codegrees[J]. Journal of Group Theory, 2020, 23(6): 1111-1122.
[37]CROOME S, LEWIS M L. Character codegrees of maximal class p-groups[J]. Canadian Mathematical Bulletin, 2020, 63(2): 328-334.
[38]MORETÓ A. Character degrees, character codegrees and nilpotence class of p-groups[J]. Communications in Algebra, 2022, 50(2): 803-808.
[39]LIU Y, YANG Y. Nonsolvable groups with four character codegrees[J]. Journal of Algebra and Its Applications, 2021, 20(11): 2150196.
[40]LIU Y, YANG Y. Nonsolvable groups with five character codegrees[J]. Communications in Algebra, 2021, 49(3): 1274-1279.
[41]BERKOVICH Y, CHILLAG D, HERZOG M. Finite groups in which the degrees of the nonlinear irreducible characters are distinct[J]. Proceedings of the American Mathematical Society, 1992, 115(4): 955-959.
[42]EBRAHIMI M. Groups in which the co degrees of the irreducible characters are distinct[J]. Communications in Algebra, 2021, 49(12): 5229-5232.
[43]BERKOVICH Y. Finite solvable groups in which only two nonlinear irreducible characters have equal degrees[J]. Journal of Algebra, 1996, 184(2): 584-603.
[44]BERKOVICH Y, KAZARIN L. Finite nonsolvable groups in which only two nonlinear irreducible characters have equal degrees[J]. Journal of Algebra, 1996, 184(2): 538-560.
[45]XIONG H. Finite groups whose character graphs associated with codegrees have no triangles[J]. Algebra Colloquium, 2016, 23(1): 15-22.
[46]SAYANJALI Z, AKHLAGHI Z, KHOSRAVI B. On the codegrees of finite groups[J]. Communications in Algebra, 2020, 48(3): 1327-1332.
[47]ALIZADEH F, GHASEMI M, GHAFFARZADEH M. Finite groups whose codegrees are almost prime[J]. Communications in Algebra, 2021, 49(2): 538-544.
[48]AHANJIDEH N. The one-prime hypothesis on the co-degrees of irreducible characters[J]. Communications in Algebra, 2021, 49(9): 4016-4020.
[49]QIAN G H. A note on element orders and character codegrees[J]. Archiv der Mathematik, 2011, 97(2): 99-103.
[50]ISSAACS I M. Element orders and character codegress[J]. Archiv der Mathematik, 2011, 97(6): 499-501.
[51]QIAN G H. Element orders and character codegrees[J]. Bulletin of the London Mathematical Society, 2021, 53(3): 820-824.
[52]GRITTINI N. p-length and character degrees in p-solvable groups[J]. Journal of Algebra, 2020, 544: 454-462.
[53]AHANJIDEH N. The fitting subgroup, p-length, derived length and character table[J]. Mathematische Nachrichten, 2021, 294(2): 214-223.
[54]YANG Y, QIAN G H. The analog of Huppert’s conjecture on character codegrees[J]. Journal of Algebra, 2017, 447: 215-219.
[55]MORETÓ A. Huppert’s conjecture for character codegrees[J]. Mathematische Nachrichten, 2021, 294(11): 2232-2236.
[56]JIN P, WANG L, YANG Y. Some counterexamples to Moretó’s question[J]. Bulletin of the Australian Mathematical Society, 已录用.
[57]BAHRI A, AKHLAGHI Z, KHOSRAVI B. An analogue of Huppert’s conjecture for character codegrees[J]. Bulletin of the Australian Mathematical Society, 2021, 104(2): 278-286.
[1] WU Xianghua, ZHONG Xianggui. Influence of NS*-permutable Subgroups on the Structure of Finite Groups [J]. Journal of Guangxi Normal University(Natural Science Edition), 2020, 38(5): 42-47.
[2] SUN Yuqing, LU Jiakuan. Influence of Self-centralizing Subgroups on the Structure of Finite Groups [J]. Journal of Guangxi Normal University(Natural Science Edition), 2020, 38(5): 48-55.
[3] LU Jiakuan,LIU Xuexia,QIN Xueqing. Notes on Frobenius Groups [J]. Journal of Guangxi Normal University(Natural Science Edition), 2018, 36(1): 84-87.
[4] ZHONG Xianggui, DING Ruifang, LING Simin. Influence of the Number of Conjugacy Classes of Nonsubnormal Subgroups on the Structure of Finite Groups [J]. Journal of Guangxi Normal University(Natural Science Edition), 2017, 35(1): 44-48.
[5] PANG Lin-na, QIU Yan-yan, LU Jia-kuan. On Some Sufficent Conditions of p-Nilpotency of Finite Groups [J]. Journal of Guangxi Normal University(Natural Science Edition), 2014, 32(2): 64-66.
[6] ZHONG Xiang-gui, LIAO Shu-hua, LI Ming-hua, TAN Chun-gui, CHEN Xiao-xiang. Maximal Supersolvable Subgroups and Supersolubility of Finite Groups [J]. Journal of Guangxi Normal University(Natural Science Edition), 2012, 30(3): 44-47.
[7] LIANG Deng-feng, YU Xin-yan, QIN Le-yang. Sovable Groups with Irreducible Character Degrees Graphs Having a Kind of Connected Components [J]. Journal of Guangxi Normal University(Natural Science Edition), 2011, 29(2): 21-25.
[8] LU Jia-kuan, MENG Wei. On π-quasinormal Subgroup of Finite Groups [J]. Journal of Guangxi Normal University(Natural Science Edition), 2011, 29(1): 35-37.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] ZHANG Xilong, HAN Meng, CHEN Zhiqiang, WU Hongxin, LI Muhang. Survey of Ensemble Classification Methods for Complex Data Stream[J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(4): 1 -21 .
[2] TONG Lingchen, LI Qiang, YUE Pengpeng. Research Progress and Prospects of Karst Soil Organic Carbon Based on CiteSpace[J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(4): 22 -34 .
[3] TIE Jun, LONG Juanjuan, ZHENG Lu, NIU Yue, SONG Yanlin. Tomato Leaf Disease Recognition Model Based on SK-EfficientNet[J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(4): 104 -114 .
[4] WENG Ye, SHAO Desheng, GAN Shu. Principal Component Liu Estimation Method of the Equation    Constrained Ⅲ-Conditioned Least Squares[J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(4): 115 -125 .
[5] QIN Chengfu, MO Fenmei. Structure ofC3-and C4-Critical Graphs[J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(4): 145 -153 .
[6] HE Qing, LIU Jian, WEI Lianfu. Single-Photon Detectors as the Physical Limit Detections of Weak Electromagnetic Signals[J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(5): 1 -23 .
[7] TIAN Ruiqian, SONG Shuxiang, LIU Zhenyu, CEN Mingcan, JIANG Pinqun, CAI Chaobo. Research Progress of Successive Approximation Register Analog-to-Digital Converter[J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(5): 24 -35 .
[8] ZHANG Shichao, LI Jiaye. Knowledge Matrix Representation[J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(5): 36 -48 .
[9] LIANG Yuting, LUO Yuling, ZHANG Shunsheng. Review on Chaotic Image Encryption Based on Compressed Sensing[J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(5): 49 -58 .
[10] HAO Yaru, DONG Li, XU Ke, LI Xianxian. Interpretability of Pre-trained Language Models: A Survey[J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(5): 59 -71 .
Copyright © Journal of Guangxi Normal University(Natural Science Edition), All Rights Reserved.
Tel: 0773-5857325　E-mail: gxsdzkb@mailbox.gxnu.edu.cn
Powered by Beijing Magtech Co. Ltd