×

zbMATH — the first resource for mathematics

Alternating and sporadic simple groups are determined by their character degrees. (English) Zbl 1252.20005
Summary: Let \(G\) be a finite group. Denote by \(\text{Irr}(G)\) the set of all irreducible complex characters of \(G\). Let \(\text{cd}(G)\) be the set of all irreducible complex character degrees of \(G\) forgetting multiplicities, that is, \(\text{cd}(G)=\{\chi(1):\chi\in\text{Irr}(G)\}\) and let \(\text{cd}^*(G)\) be the set of all irreducible complex character degrees of \(G\) counting multiplicities. Let \(H\) be an alternating group of degree at least 5, a sporadic simple group or the Tits group. In this paper, we show that if \(G\) is a non-Abelian simple group and \(\text{cd}(G)\subseteq\text{cd}(H)\) then \(G\) must be isomorphic to \(H\). As a consequence, we show that if \(G\) is a finite group with \(\text{cd}^*(G)\subseteq\text{cd}^*(H)\) then \(G\) is isomorphic to \(H\). This gives a positive answer to Question 11.8(a) in The Kourovka notebook. Unsolved problems in group theory [16th ed. (2006; Zbl 1084.20001)] for alternating groups, sporadic simple groups or the Tits group.

MSC:
20C15 Ordinary representations and characters
20C30 Representations of finite symmetric groups
20C34 Representations of sporadic groups
20D06 Simple groups: alternating groups and groups of Lie type
20D08 Simple groups: sporadic groups
Software:
GAP
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Balog, A., Bessenrodt, C., Olsson, J., Ono, K.: Prime power degree representations of the symmetric and alternating groups. J. Lond. Math. Soc. (2) 64(2), 344–356 (2001) · Zbl 1018.20008
[2] Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, Eynsham (1985) · Zbl 0568.20001
[3] Hagie, M.: The prime graph of a sporadic simple group. Commun. Algebra 31(9), 4405–4424 (2003) · Zbl 1031.20009
[4] Herzog, M.: On finite simple groups of order divisible by three primes only. J. Algebra 10, 383–388 (1968) · Zbl 0167.29101
[5] Huppert, B.: Some simple groups which are determined by the set of their character degrees. I. Ill. J. Math. 44(4), 828–842 (2000) · Zbl 0972.20006
[6] Huppert, B.: Some simple groups which are determined by the set of their character degrees. II. Rend. Semin. Mat. Univ. Padova 115, 1–13 (2006) · Zbl 1156.20008
[7] Isaacs, M. Character Theory of Finite Groups. Corrected Reprint of the 1976 Original, Academic Press, New York. AMS Chelsea Publishing, Providence, RI (2006)
[8] James, G.: On the minimal dimensions of irreducible representations of symmetric groups. Math. Proc. Camb. Philos. Soc. 94(3), 417–424 (1983) · Zbl 0544.20011
[9] Lübeck, F.: Smallest degrees of representations of exceptional groups of Lie type. Commun. Algebra 29(5), 2147–2169 (2001) · Zbl 1004.20003
[10] Lusztig, G.: On the representations of reductive groups with disconnected centre. Orbites unipotentes et représentations, I. Astérisque 10(168), 157–166 (1988)
[11] Malle, G.: Almost irreducible tensor squares. Commun. Algebra 27(3), 1033–1051 (1999) · Zbl 0931.20009
[12] Malle, G., Zalesskii, A.: Prime power degree representations of quasi-simple groups. Arch. Math. (Basel) 77(6), 461–468 (2001) · Zbl 0996.20006
[13] Manz, O., Staszewski, R., Willems, W.: On the number of components of a graph related to character degrees. Proc. Am. Math. Soc. 103(1), 31–37 (1988) · Zbl 0645.20005
[14] Michler, G.: Brauer’s conjectures and the classification of finite simple groups. Representation Theory, II (Ottawa, Ont., 1984). Lecture Notes in Math. pp. 129–142, vol. 1178. Springer, Berlin (1986)
[15] Mazurov, V.D., Khukhro, E.I. (eds.): Unsolved Problems in Group Theory, The Kourovka Notebook, No. 16. Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk (2006) · Zbl 1084.20001
[16] Rasala, R.: On the minimal degrees of characters of S n . J. Algebra 45(1), 132–181 (1977) · Zbl 0348.20009
[17] Roger Carter, W.: Finite Groups of Lie Type. Conjugacy Classes and Complex Characters. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. Wiley, New York (1985) · Zbl 0567.20023
[18] The GAP Group, GAP–Groups: Algorithms, and Programming, Version 4.4.10, http://www.gap-system.org (2007)
[19] Wakefield, T.: Verifying Huppert’s conjecture for ${}\^2G_2(q\^2)$ . Algebr. Represent. Theory. doi: 10.1007/s10468-009-9206-x (2009)
[20] Wakefield, T.: Verifying Huppert’s conjecture for PSL 3(q) and $PSU_3(q\^2)$ . Commun. Algebra 37(8), 2887–2906 (2009) · Zbl 1185.20014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.