Huppert and Qian et al. classified finite groups for which all irreducible character degrees are consecutive. The aim of this paper is to determine the structure of finite groups whose irreducible character degrees of their proper subgroups are all consecutive.
Citation: Shitian Liu. Finite groups for which all proper subgroups have consecutive character degrees[J]. AIMS Mathematics, 2023, 8(3): 5745-5762. doi: 10.3934/math.2023289
Huppert and Qian et al. classified finite groups for which all irreducible character degrees are consecutive. The aim of this paper is to determine the structure of finite groups whose irreducible character degrees of their proper subgroups are all consecutive.
[1] | H. R. Brahana, On the metabelian groups which contain a given group H as a maximal invariant abelian subgroup, Amer. J. Math., 56 (1934), 490–510. https://doi.org/10.2307/2370950 doi: 10.2307/2370950 |
[2] | R. Brandl, Groups with few non-nilpotent subgroups, J. Algebra Appl., 16 (2017), 1750188. https://doi.org/10.1142/S0219498817501882 doi: 10.1142/S0219498817501882 |
[3] | J. N. Bray, D. F. Holt, C. M. Roney-Dougal, The maximal subgroups of the low-dimensional finite classical groups, In: London mathematical society lecture note series, Cambridge: Cambridge University Press, 2013. https://doi.org/10.1017/CBO9781139192576 |
[4] | T. Breuer, The GAP character table lbrary, 2012. http://www.math.rwth-aachen.de/Thomas.Breuer/ctbllib. |
[5] | S. A. Chen, Finite solvable groups with irreducible character degrees of arithmetic number, Acta Math. Sinica Chin. Ser., 56 (2013), 31–40. |
[6] | Z. M. Chen, Inner- and outer-supersolvable groups, and sufficient conditions for a group to be supersolvable, Acta Math. Sinica Chin. Ser., 27 (1984), 694–703. |
[7] | J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Eynsham: Oxford University Press, 1985. |
[8] | L. E. Dickson, Linear groups: with an exposition of the Galois field theory, New York: Dover Publications, 1958. |
[9] | L. Dornhoff, Group representation theory. Part A: Ordinary representation theory, New York: Marcel Dekker, 1971. |
[10] | M. Geck, G. Malle, The character theory of finite groups of Lie type, In: Cambridge studies in advanced mathematics, Cambridge: Cambridge University Press, 2020. |
[11] | M. Giudici, Maximal subgroups of almost simple groups with socle $PSL(2, q)$, arXiv: Group Theory, 2007, 0703685. https://doi.org/10.48550/arXiv.math/0703685 |
[12] | B. Huppert, Die grundlehren der mathematischen wissenschaften, band 134, In: Endliche gruppen. I, Berlin: Springer, 1967. |
[13] | B. Huppert, A characterization of GL(2, 3) and SL(2, 5) by the degrees of their representations, Forum Math., 1 (1989), 167–183. https://doi.org/10.1515/form.1989.1.167 doi: 10.1515/form.1989.1.167 |
[14] | B. Huppert, Character theory of finite groups, In: De gruyter expositions in mathematics, Berlin: Walter de Gruyter & Co., 1998. https://doi.org/10.1515/9783110809237 |
[15] | I. M. Isaacs, Character theory of finite groups, New York: Dover Publications, 1994. |
[16] | H. E. Jordan, Group-characters of various types of linear groups, American J. Math., 29 (1907), 387–405. https://doi.org/10.2307/2370015 doi: 10.2307/2370015 |
[17] | P. B. Kleidman, The subgroup structure of some finite simple groups, Ph.D Thesis, University of Cambridge, 1987. |
[18] | Q. Li, X. Guo, On generalization of minimal non-nilpotent groups, Lobachevskii J. Math., 31 (2010), 239–243. https://doi.org/10.1134/S1995080210030078 doi: 10.1134/S1995080210030078 |
[19] | M. W. Liebeck, J. Saxl, On the orders of maximal subgroups of the finite exceptional groups of Lie type, Proc. London Math. Soc., 55 (1987), 299–330. |
[20] | S. Liu, On groups whose irreducible character degrees of all proper subgroups are all prime powers, J. Math., 2021 (2021), 6345386. https://doi.org/10.1155/2021/6345386 doi: 10.1155/2021/6345386 |
[21] | S. Liu, D. Lei, X. Li, On groups with consecutive three smallest character degrees, ScienceAsia, 45 (2019), 474–481. |
[22] | S. Liu, X. Tang, Nonsolvable groups whose degrees of all proper subgroups are the direct products of at most two prime numbers, J. Math., 2022 (2022), 1455299. https://doi.org/10.1155/2022/1455299 doi: 10.1155/2022/1455299 |
[23] | Rwth-Aachen University, Character degrees and their multiplicities for some groups of lie type of rank < 9, 2007. Available from: http://www.math.rwth-aachen.de/ Frank.Luebeck/chev/DegMult/index.html?LANG=en. |
[24] | G. Malle, The maximal subgroups of ${}^2F_4(q^2)$, J. Algebra, 139 (1991), 52–69. https://doi.org/10.1016/0021-8693(91)90283-E doi: 10.1016/0021-8693(91)90283-E |
[25] | O. Manz, W. Willems, T. R. Wolf, The diameter of the character degree graph, J. Reine Angew. Math., 402 (1989), 181–198. |
[26] | G. Qian, Finite groups with consecutive nonlinear character degrees, J. Algebra, 285 (2005), 372–382. https://doi.org/10.1016/j.jalgebra.2004.11.021 doi: 10.1016/j.jalgebra.2004.11.021 |
[27] | J. W. Randolph, Finite groups with solvable maximal subgroups, Proc. Amer. Math. Soc., 23 (1969), 490–492. https://doi.org/10.2307/2036570 doi: 10.2307/2036570 |
[28] | J. Sangroniz, Character degrees of the Sylow $p$-subgroups of classical groups, Cambridge: Cambridge University Press, 2003. https://doi.org/10.1017/CBO9780511542787.016 |
[29] | W. A. Simpson, J. S. Frame, The character tables for SL$(3, q)$, SU$(3, q^{2})$, PSL$(3, q)$, PSU$(3, q^{2})$, Canad. J. Math., 25 (1973), 486–494. https://doi.org/10.4153/CJM-1973-049-7 doi: 10.4153/CJM-1973-049-7 |
[30] | R. Steinberg, The representations of GL$(3, q)$, GL$(4, q)$, PGL$(3, q)$, and PGL$(4, q)$, Canad. J. Math., 3 (1951), 225–235. https://doi.org/10.4153/cjm-1951-027-x doi: 10.4153/cjm-1951-027-x |
[31] | J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc., 74 (1968), 383–437. https://doi.org/10.1090/S0002-9904-1968-11953-6 doi: 10.1090/S0002-9904-1968-11953-6 |
[32] | P. H. Tiep, A. E. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra, 24 (1996), 2093–2167. https://doi.org/10.1080/00927879608825690 doi: 10.1080/00927879608825690 |
[33] | L. Wang, On the automorphism groups of Frobenius groups, Comm. Algebra, 48 (2020), 5330–5342. https://doi.org/10.1080/00927872.2020.1788045 doi: 10.1080/00927872.2020.1788045 |
[34] | H. N. Ward, On Ree's series of simple groups, Trans. Amer. Math. Soc., 121 (1966), 62–89. https://doi.org/10.2307/1994333 doi: 10.2307/1994333 |
[35] | D. L. White, Character degrees of extensions of PSL$_2(q)$ and SL$_2(q)$, J. Group Theory, 16 (2013), 1–33. https://doi.org/10.1515/jgt-2012-0026 doi: 10.1515/jgt-2012-0026 |
[36] | H. Xu, G. Chen, Y. Yan, A new characterization of simple $K_3$-groups by their orders and large degrees of their irreducible characters, Comm. Algebra, 42 (2014), 5374–5380. https://doi.org/10.1080/00927872.2013.842242 doi: 10.1080/00927872.2013.842242 |