Let
be a finite group and let
be the sum of codegrees of complex irreducible characters of a group
. Say
. We prove that a finite group
such that for all proper subgroup
,
, is solvable.
References
[1]
Akhlaghi, Z. (2024) On the Average Degree of Linear and Even Degree Characters of Finite Groups. Ricerche di Matematica, 73, 2385-2395. https://doi.org/10.1007/s11587-022-00750-0
[2]
Eskandari, E. and Ahanjideh, N. (2023) On p-Solvability and Average Character Degree in a Finite Group. Bulletin of the Australian Mathematical Society, 109, 507-511. https://doi.org/10.1017/s0004972723000722
[3]
Herzog, M., Longobardi, P. and Maj, M. (2023) On Groups with Average Element Orders Equal to the Average Order of the Alternating Group of Degree 5. GlasnikMatematicki, 58, 307-315. https://doi.org/10.3336/gm.58.2.10
[4]
Moretó, A. (2023) The Average Character Degree of Finite Groups and Gluck’s Conjecture. Journal of Group Theory, 26, 803-815. https://doi.org/10.1515/jgth-2022-0120
[5]
Qian, G., Wang, Y. and Wei, H. (2007) Co-Degrees of Irreducible Characters in Finite Groups. Journal of Algebra, 312, 946-955. https://doi.org/10.1016/j.jalgebra.2006.11.001
[6]
Gintz, M., Kortje, M., Laurence, M., Liu, Y., Wang, Z. and Yang, Y. (2022) On the Characterization of Some Non-Abelian Simple Groups with Few Codegrees. Communications in Algebra, 50, 3932-3939. https://doi.org/10.1080/00927872.2022.2049807
[7]
Hung, N.N. and Moretó, A. (2025) The Codegree Isomorphism Problem for Finite Simple Groups II. The Quarterly Journal of Mathematics, 76, 237-250. https://doi.org/10.1093/qmath/haaf001
[8]
Lewis, M.L. and Yan, Q. (2025) On the Sum of Character Codegrees of Finite Groups. Monatshefte für Mathematik, 206, 143-160. https://doi.org/10.1007/s00605-024-02033-2
[9]
Li, P. and Qu, H. (2024) Finite p-Groups with Three Character Codegrees. Communications in Algebra, 52, 4149-4154. https://doi.org/10.1080/00927872.2024.2342544
[10]
Liu, S. (2023) Finite Groups Whose Numbers of Real-Valued Character Degrees of All Proper Subgroups Are at Most Two. Proceedings of the Bulgarian Academy of Sciences, 76, 990-998. https://doi.org/10.7546/crabs.2023.07.02
[11]
Qian, G. (2025) Finite Groups with Non-Complete Character Codegree Graphs. Journal of Algebra, 669, 75-94. https://doi.org/10.1016/j.jalgebra.2025.01.022
[12]
Wang, Z., Qian, G., Lv, H. and Chen, G. (2023) On the Average Codegree of a Finite Group. Journal of Algebra and Its Applications, 23, Article 2450102. https://doi.org/10.1142/s0219498824501020
[13]
Liu, S. (2022) Finite Groups for Which All Proper Subgroups Have Consecutive Character Degrees. AIMS Mathematics, 8, 5745-5762. https://doi.org/10.3934/math.2023289
[14]
Liu, S. and Tang, X. (2022) Nonsolvable Groups Whose Degrees of All Proper Subgroups Are the Direct Products of at Most Two Prime Numbers. Journal of Mathematics, 2022, Article 1455299. https://doi.org/10.1155/2022/1455299
[15]
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A. and Wilson, R.A. (1985) Atlas of Finite Groups. Oxford University Press.
[16]
Breuer, T. (2012) The GAP Character Table Library, Version 1.3.9. http://www.math.rwth-aachen.de/homes/Thomas.Breuer/ctbllib
[17]
Isaacs, I.M. (1994) Character Theory of Finite Groups. Dover Publications, Inc.
[18]
Jordan, H.E. (1907) Group-Characters of Various Types of Linear Groups. American Journal of Mathematics, 29, 387-405. https://doi.org/10.2307/2370015
[19]
Thompson, J.G. (1968) Nonsolvable Finite Groups All of Whose Local Subgroups Are Solvable. Bulletin of the American Mathematical Society, 74, 383-437. https://doi.org/10.1090/s0002-9904-1968-11953-6
[20]
James, G. and Liebeck, M. (2001) Representations and Characters of Groups. 2nd Edition, Cambridge University Press. https://doi.org/10.1017/cbo9780511814532
[21]
Huppert, B. (1967) Endliche Gruppen. I. Die grundlehren der mathematischen wissenschaften. Vol. 134, Springer-Verlag.
[22]
Bray, J.N., Holt, D.F. and Roney-Dougal, C.M. (2013) The Maximal Subgroups of the Low-Dimensional Finite Classical Groups. Cambridge University Press. https://doi.org/10.1017/cbo9781139192576
