# zbMATH — the first resource for mathematics

A new characterization for the simple group $$\mathrm{PSL}(2,p^2)$$ by order and some character degrees. (English) Zbl 1363.20031
For a finite group $$G$$, let $$X_1(G)$$ denote the set of irreducible complex character degrees of $$G$$, counting multiplicitites, i.e. the first column of the character table of $$G$$. It is a well-known fact from basic representation theory that $$X_1(G)$$ completely determines the group algebra $$\mathbb C G$$. This means that $$X_1(G)=X_1(H)$$ for two finite groups $$G$$ and $$H$$ if and only if $$\mathbb C G\simeq \mathbb C H$$.
There are examples, when the isomorphism $$\mathbb C G\simeq \mathbb C H$$ does not imply that $$G\simeq H$$. In contrast, H. P. Tong-Viet [Monatsh. Math. 166, No. 3–4, 559–577 (2012; Zbl 1255.20006); Algebr. Represent. Theory 15, No. 2, 379–389 (2012; Zbl 1252.20005); J. Algebra 357, 61–68 (2012; Zbl 1259.20008)] proved recently that if $$S$$ is a finite simple group and $$G$$ is any finite group such that $$X_1(G)=X_1(S)$$ (or, equivalently, $$\mathbb C G\simeq \mathbb C S$$), then $$G\simeq S$$ must hold.
In this paper, the authors strengthen this result for the case $$S\simeq \mathrm{PSL}(2,p^2)$$, where $$p$$ is an odd prime. Their main theorem says that if $$G$$ is a finite group such that $$|G|=|\mathrm{PSL}(2,p^2)|,\;p^2\in X_1(G)$$ but no element of $$X_1(G)$$ is divisible by $$2p$$, then $$G\simeq \mathrm{PSL}(2,p^2)$$.

##### MSC:
 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D06 Simple groups: alternating groups and groups of Lie type 20G40 Linear algebraic groups over finite fields 20C15 Ordinary representations and characters 20D05 Finite simple groups and their classification
##### Keywords:
character degree; order; projective special linear group
Full Text:
##### References:
  J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon Press, Oxford, 1985. · Zbl 0568.20001  P. Crescenzo: A diophantine equation which arises in the theory of finite groups. Adv. Math. 17 (1975), 25–29. · Zbl 0305.10016  B. Huppert: Some simple groups which are determined by the set of their character degrees. I. Ill. J. Math. 44 (2000), 828–842. · Zbl 0972.20006  B. Huppert: Character Theory of Finite Groups. De Gruyter Expositions in Mathematics 25, Walter de Gruyter, Berlin, 1998. · Zbl 0932.20007  I. M. Isaacs: Character degree graphs and normal subgroups. Trans. Am. Math. Soc. 356 (2004), 1155–1183. · Zbl 1034.20009  I. M. Isaacs: Character Theory of Finite Groups. Pure and Applied Mathematics 69, Academic Press, New York, 1976.  B. Khosravi: Groups with the same orders and large character degrees as PGL(2, 9). Quasigroups Relat. Syst. 21 (2013), 239–243. · Zbl 1294.20009  B. Khosravi, B. Khosravi, B. Khosravi: Recognition of PSL(2, p) by order and some information on its character degrees where p is a prime. Monatsh. Math. 175 (2014), 277–282. · Zbl 1304.20042  W. Kimmerle: Group rings of finite simple groups. Resen. Inst. Mat. Estat. Univ. São Paulo 5 (2002), 261–278. · Zbl 1047.20007  M. L. Lewis, D. L. White: Nonsolvable groups with no prime dividing three character degrees. J. Algebra 336 (2011), 158–183. · Zbl 1246.20006  M. Nagl: Charakterisierung der symmetrischen Gruppen durch ihre komplexe Gruppenalgebra. Stuttgarter Mathematische Berichte, http://www.mathematik.uni-stuttgart.de/preprints/downloads/2011/2011-007.pdf (2011). (In German.)  M. Nagl: Über das Isomorphieproblem von Gruppenalgebren endlicher einfacher Gruppen. Diplomarbeit, Universität Stuttgart, 2008. (In German.)  H. P. Tong-Viet: Alternating and sporadic simple groups are determined by their character degrees. Algebr. Represent. Theory 15 (2012), 379–389. · Zbl 1252.20005  H. P. Tong-Viet: Simple classical groups of Lie type are determined by their character degrees. J. Algebra 357 (2012), 61–68. · Zbl 1259.20008  H. P. Tong-Viet: Simple exceptional groups of Lie type are determined by their character degrees. Monatsh. Math. 166 (2012), 559–577. · Zbl 1255.20006  H. P. Tong-Viet: Symmetric groups are determined by their character degrees. J. Algebra 334 (2011), 275–284. · Zbl 1246.20007  D. L. White: Degree graphs of simple groups. Rocky Mt. J. Math. 39 (2009), 1713–1739. · Zbl 1180.20008  H. Xu, G. Chen, Y. Yan: A new characterization of simple K3-groups by their orders and large degrees of their irreducible characters. Commun. Algebra 42 (2014), 5374–5380. · 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.