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Keywords:
character degree; order; projective special linear group
character degree; order; projective special linear group
Summary:
Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $|{\rm PSL}(2,p^2)|$ such that $G$ has an irreducible character of degree $p^2$ and we know that $G$ has no irreducible character $\theta $ such that $2p\mid \theta (1)$, then $G$ is isomorphic to ${\rm PSL}(2,p^2)$. As a consequence of our result we prove that ${\rm PSL}(2,p^2)$ is uniquely determined by the structure of its complex group algebra.
References:
[1] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A.: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon Press, Oxford (1985). MR 0827219 | Zbl 0568.20001
[2] Crescenzo, P.: A diophantine equation which arises in the theory of finite groups. Adv. Math. 17 (1975), 25-29. DOI 10.1016/0001-8708(75)90083-3 | MR 0371812 | Zbl 0305.10016
[3] Huppert, B.: Some simple groups which are determined by the set of their character degrees. I. Ill. J. Math. 44 (2000), 828-842. MR 1804317 | Zbl 0972.20006
[4] Huppert, B.: Character Theory of Finite Groups. De Gruyter Expositions in Mathematics 25 Walter de Gruyter, Berlin (1998). MR 1645304 | Zbl 0932.20007
[5] Isaacs, I. M.: Character degree graphs and normal subgroups. Trans. Am. Math. Soc. 356 (2004), 1155-1183. DOI 10.1090/S0002-9947-03-03462-7 | MR 2021616 | Zbl 1034.20009
[6] Isaacs, I. M.: Character Theory of Finite Groups. Pure and Applied Mathematics 69 Academic Press, New York (1976). MR 0460423 | Zbl 0337.20005
[7] Khosravi, B.: Groups with the same orders and large character degrees as $ PGL(2,9)$. Quasigroups Relat. Syst. 21 (2013), 239-243. MR 3203150 | Zbl 1294.20009
[8] Khosravi, B., Khosravi, B., Khosravi, B.: 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. DOI 10.1007/s00605-013-0582-2 | MR 3260870 | Zbl 1304.20042
[9] Kimmerle, W.: Group rings of finite simple groups. Resen. Inst. Mat. Estat. Univ. São Paulo 5 (2002), 261-278. MR 2015338 | Zbl 1047.20007
[10] Lewis, M. L., White, D. L.: Nonsolvable groups with no prime dividing three character degrees. J. Algebra 336 (2011), 158-183. DOI 10.1016/j.jalgebra.2011.03.028 | MR 2802535 | Zbl 1246.20006
[11] Nagl, M.: Charakterisierung der symmetrischen Gruppen durch ihre komplexe Gruppenalgebra. Stuttgarter Mathematische Berichte, http://www.mathematik.uni-stuttgart. de/preprints/downloads/2011/2011-007.pdf (2011), German.
[12] Nagl, M.: Über das Isomorphieproblem von Gruppenalgebren endlicher einfacher Gruppen. Diplomarbeit, Universität Stuttgart (2008), German.
[13] Tong-Viet, H. P.: Alternating and sporadic simple groups are determined by their character degrees. Algebr. Represent. Theory 15 (2012), 379-389. DOI 10.1007/s10468-010-9247-1 | MR 2892513 | Zbl 1252.20005
[14] Tong-Viet, H. P.: Simple classical groups of Lie type are determined by their character degrees. J. Algebra 357 (2012), 61-68. DOI 10.1016/j.jalgebra.2012.02.011 | MR 2905242 | Zbl 1259.20008
[15] Tong-Viet, H. P.: Simple exceptional groups of Lie type are determined by their character degrees. Monatsh. Math. 166 (2012), 559-577. DOI 10.1007/s00605-011-0301-9 | MR 2925155 | Zbl 1255.20006
[16] Tong-Viet, H. P.: Symmetric groups are determined by their character degrees. J. Algebra 334 (2011), 275-284. DOI 10.1016/j.jalgebra.2010.11.018 | MR 2787664 | Zbl 1246.20007
[17] White, D. L.: Degree graphs of simple groups. Rocky Mt. J. Math. 39 (2009), 1713-1739. DOI 10.1216/RMJ-2009-39-5-1713 | MR 2546661 | Zbl 1180.20008
[18] Xu, H., Chen, G., Yan, Y.: A new characterization of simple $K_3$-groups by their orders and large degrees of their irreducible characters. Commun. Algebra 42 5374-5380 (2014). DOI 10.1080/00927872.2013.842242 | MR 3223645 | Zbl 1297.20012
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