zbMATH — the first resource for mathematics

Recognition of \(\text{PSL}(2,p)\) by order and some information on its character degrees where \(p\) is a prime. (English) Zbl 1304.20042
Summary: Let \(G\) be a finite group and \(\text{cd}(G)\) be the set of irreducible character degrees of \(G\). In this paper we prove that if \(p\) is a prime number, then the simple group \(\text{PSL}(2,p)\) is uniquely determined by its order and some information about its character degrees. In fact we prove that if \(G\) is a finite group such that (i) \(|G|=|\text{PSL}(2,p)|\), (ii) \(p\in\text{cd}(G)\), (iii) \(\text{cd}(G)\) has an even integer, and (iv) there does not exist any element \(a\in\text{cd}(G)\) such that \(2p\mid a\), then \(G\cong\text{PSL}(2,p)\). As a consequence of our result we get that \(\text{PSL}(2,p)\) is uniquely determined by its order and the largest and the second largest character degrees.

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20C15 Ordinary representations and characters
20D06 Simple groups: alternating groups and groups of Lie type
20C33 Representations of finite groups of Lie type
Full Text: DOI
[1] Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, Oxford (1985) · Zbl 0568.20001
[2] Huppert, B.: Character Theory of Finite Groups. de Gruyter, Berlin (1998) · Zbl 0932.20007
[3] Huppert, B, Some simple groups which are determined by the set of their character degrees (I), Ill. J. Math., 44, 828-842, (2000) · Zbl 0972.20006
[4] Isaacs, I.M.: Character Theory of Finite Groups. Academic Press, New York (1976) · Zbl 0337.20005
[5] Isaacs, IM, Character degree graphs and normal subgroups, Trans. Am. Math. Soc., 356, 1155-1183, (2004) · Zbl 1034.20009
[6] Khosravi, B.: Groups with the same order and large character degrees as \({\rm PGL(2,9)}\), Quasigr. Relat. Syst. (to appear)
[7] Lewis, ML; White, DL, Nonsolvable groups with no prime dividing three character degrees, J. Algebra, 336, 158-183, (2011) · Zbl 1246.20006
[8] Xu, H., Chen, G.Y., Yan, Y.: A new characterization of simple \(K_3\)-groups by their orders and large degrees of their irreducible characters. Comm. Algebra (to appear) · Zbl 1297.20012
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.