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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
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##### References:
 [1] 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 [2] P. Crescenzo: A diophantine equation which arises in the theory of finite groups. Adv. Math. 17 (1975), 25–29. · Zbl 0305.10016 [3] 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 [4] B. Huppert: Character Theory of Finite Groups. De Gruyter Expositions in Mathematics 25, Walter de Gruyter, Berlin, 1998. · Zbl 0932.20007 [5] I. M. Isaacs: Character degree graphs and normal subgroups. Trans. Am. Math. Soc. 356 (2004), 1155–1183. · Zbl 1034.20009 [6] I. M. Isaacs: Character Theory of Finite Groups. Pure and Applied Mathematics 69, Academic Press, New York, 1976. [7] 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 [8] 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 [9] W. Kimmerle: Group rings of finite simple groups. Resen. Inst. Mat. Estat. Univ. São Paulo 5 (2002), 261–278. · Zbl 1047.20007 [10] M. L. Lewis, D. L. White: Nonsolvable groups with no prime dividing three character degrees. J. Algebra 336 (2011), 158–183. · Zbl 1246.20006 [11] 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.) [12] M. Nagl: Über das Isomorphieproblem von Gruppenalgebren endlicher einfacher Gruppen. Diplomarbeit, Universität Stuttgart, 2008. (In German.) [13] 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 [14] 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 [15] 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 [16] H. P. Tong-Viet: Symmetric groups are determined by their character degrees. J. Algebra 334 (2011), 275–284. · Zbl 1246.20007 [17] D. L. White: Degree graphs of simple groups. Rocky Mt. J. Math. 39 (2009), 1713–1739. · Zbl 1180.20008 [18] 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
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