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Keywords:
character degree; complex group algebra; projective general unitary group
character degree; complex group algebra; projective general unitary group
Summary:
Let $G$ be a finite group. Let $X_1(G)$ be the first column of the ordinary character table of $G$. We will show that if $X_1(G)=X_1({\rm PGU}_3(q^2))$, then $G \cong {\rm PGU}_3(q^2)$. As a consequence, we show that the projective general unitary groups ${\rm PGU}_3(q^2)$ are uniquely determined by the structure of their complex group algebras.
References:
[1] Berkovich, Y. G., Zhmud', E. M.: Characters of Finite Groups. Part 1. Translations of Mathematical Monographs 172, American Mathematical Society, Providence (1998). MR 1486039 | Zbl 0934.20008
[2] Bessenrodt, C., Nguyen, H. N., Olsson, J. B., Tong-Viet, H. P.: Complex group algebras of the double covers of the symmetric and alternating groups. Algebra Number Theory 9 (2015), 601-628. DOI 10.2140/ant.2015.9.601 | MR 3340546 | Zbl 1321.20011
[3] Brauer, R.: Representations of finite groups. Lect. Modern Math. 1 (1963), 133-175. MR 0178056 | Zbl 0124.26504
[4] Carter, R. W.: Finite Groups of Lie Type. Conjugacy Classes and Complex Characters. Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Chichester (1993). MR 1266626 | Zbl 0567.20023
[5] Dolfi, S., Navarro, G., Tiep, P. H.: Primes dividing the degrees of the real characters. Math. Z. 259 (2008), 755-774. DOI 10.1007/s00209-007-0247-8 | MR 2403740 | Zbl 1149.20006
[6] Gorenstein, D., Lyons, R., Solomon, R.: The Classification of the Finite Simple Groups. Part I. Chapter A: Almost Simple $K$-Groups. Mathematical Surveys and Monographs 40, American Mathematical Society, Providence (1998). MR 1490581 | Zbl 0890.20012
[7] Heydari, S., Ahanjideh, N.: A characterization of $ PGL(2,p^n)$ by some irreducible complex character degrees. Publ. Inst. Math., Nouv. Sér. 99 (2016), 257-264. DOI 10.2298/PIM150111017H | MR 3524049 | Zbl 06749629
[8] Huppert, B.: Some simple groups, which are determined by the set of their character degrees III. Preprint. MR 1804317
[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.: Solvable groups whose degree graphs have two connected components. J. Group Theory 4 (2001), 255-275. DOI 10.1515/jgth.2001.023 | MR 1839998 | Zbl 0998.20009
[11] Lübeck, F.: Data for finite groups of Lie type and related algebraic groups. Available at http://www.math.rwth-aachen.de/$\sim$Frank.Luebeck/chev/index.html
[12] Malle, G., Testerman, D.: Linear Algebraic Groups and Finite groups of Lie Type. Cambridge Studies in Advanced Mathematics 133, Cambridge University Press, Cambridge (2011). DOI 10.1017/CBO9780511994777 | MR 2850737 | Zbl 1256.20045
[13] Meng, Q., Zeng, J.: Finite groups whose character degree graphs are empty graphs. Algebra Colloq. 20 (2013), 75-80. DOI 10.1142/S1005386713000060 | MR 3020718 | Zbl 1280.20009
[14] Nagl, M.: Charakterisierung der Symmetrischen Gruppen durch ihre komplexe Gruppenalgebra. Stuttgarter Mathematische Berichte (2011), German Available at http://www.mathematik.uni-stuttgart.de/preprints/downloads/2011/2011-007.pdf
[15] Nguyen, H. N.: Quasisimple classical groups and their complex group algebras. Isr. J. Math. 195 (2013), 973-998. DOI 10.1007/s11856-012-0142-9 | MR 3096579 | Zbl 1294.20004
[16] Nguyen, H. N., Tong-Viet, H. P.: Characterizing finite quasisimple groups by their complex group algebras. Algebr. Represent. Theory 17 (2014), 305-320. DOI 10.1007/s10468-012-9400-0 | MR 3160726 | Zbl 1303.20001
[17] Shirjian, F., Iranmanesh, A.: Complex group algebras of almost simple groups with socle $PSL_n(q)$. To appear in Commun. Algebra. DOI 10.1080/00927872.2017.1324868
[18] Simpson, W. A., Frame, J. S.: The character tables for $ SL(3,q)$, $ SU(3,q^2)$, $ PSL(3,q)$, $ PSU(3,q^2)$. Can. J. Math. 25 (1973), 486-494. DOI 10.4153/CJM-1973-049-7 | MR 0335618 | Zbl 0264.20010
[19] 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
[20] 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
[21] 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
[22] 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
[23] Wakefield, T. P.: Verifying Huppert's conjecture for $ PSL_3(q)$ and $ PSU_3(q^2)$. Commun. Algebra 37 (2009), 2887-2906. DOI 10.1080/00927870802625661 | MR 2543522 | Zbl 1185.20014
[24] Zsigmondy, K.: On the theory of power residues. Monatsh. Math. Phys. 3 (1892), 265-284 German \99999JFM99999 24.0176.02. DOI 10.1007/BF01692444 | MR 1546236
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