Abstract
The spectrum of a finite group is the set of the orders of its elements. We consider the problem that arises within the framework of recognition of finite simple groups by spectrum: Determine all finite almost simple groups having the same spectrum as its socle. This problem was solved for all almost simple groups with exception of the case that the socle is a simple even-dimensional orthogonal group over a field of odd characteristic. Here we address this remaining case and determine the almost simple groups in question.
Also we prove that there are infinitely many pairwise nonisomorphic finite groups having the same spectrum as the simple 8-dimensional symplectic group over a field of characteristic other than 7.
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Zavarnitsin A. V. and Mazurov V. D., “Element orders in coverings of symmetric and alternating groups,” Algebra and Logic, vol. 38, No. 3, 159–170 (1999).
Zavarnitsine A. V., “Properties of element orders in covers for Ln(q) and Un(q),” Sib. Math. J., vol. 49, No. 2, 246–256 (2008).
Vasilev A. V., Grechkoseeva M. A., and Mazurov V. D., “On finite groups isospectral to simple symplectic and orthogonal groups,” Sib. Math. J., vol. 50, No. 6, 965–981 (2009).
Vasilev A. V., Grechkoseeva M. A., and Staroletov A. M., “On finite groups isospectral to simple linear and unitary groups,” Sib. Math. J., vol. 52, No. 1, 30–40 (2011).
Grechkoseeva M. A., “On element orders in covers of finite simple classical groups,” J. Algebra, vol. 339, 304–319 (2011).
Gorshkov I. B., “Recognizability of alternating groups by spectrum,” Algebra and Logic, vol. 52, No. 1, 41–45 (2013).
Grechkoseeva M. A., “On element orders in covers of finite simple groups of Lie type,” J. Algebra Appl., vol. 14, 1550056 (2015).
Vasilev A. V., “On finite groups isospectral to simple classical groups,” J. Algebra, vol. 423, 318–374 (2015).
Grechkoseeva M. A. and Vasilev A. V., “On the structure of finite groups isospectral to finite simple groups,” J. Group Theory, vol. 18, No. 5, 741–759 (2015).
Vasilev A. V. and Staroletov A. M., “Almost recognizability by spectrum of simple exceptional groups of Lie type,” Algebra and Logic, vol. 53, No. 6, 433–449 (2014).
Vasilev A. V. and Grechkoseeva M. A., “Recognition by spectrum for simple classical groups in characteristic 2,” Sib. Math. J., vol. 56, No. 6, 1009–1018 (2015).
Staroletov A., “On almost recognizability by spectrum of simple classical groups,” Int. J. Group Theory, vol. 6, No. 4, 7–33 (2017).
Grechkoseeva M. A., “Recognition by spectrum for finite linear groups over fields of characteristic 2,” Algebra and Logic, vol. 47, No. 4, 229–241 (2008).
Grechkoseeva M. A. and Shi W. J., “On finite groups isospectral to finite simple unitary groups over fields of characteristic 2,” Sib. Elektron. Mat. Izv., vol. 10, 31–37 (2013).
Grechkoseeva M. A., “On orders of elements of finite almost simple groups with linear or unitary socle,” J. Group Theory, vol. 20, No. 6, 1191–1222 (2017).
Zvezdina M. A., “Spectra of automorphic extensions of finite simple symplectic and orthogonal groups over fields of characteristic 2,” Sib. Èlektron. Mat. Izv., vol. 11, 823–832 (2014).
Grechkoseeva M. A., “On spectra of almost simple groups with symplectic or orthogonal socle,” Sib. Math. J., vol. 57, No. 4, 582–587 (2016).
Zvezdina M. A., “Spectra of automorphic extensions of finite simple exceptional groups of Lie type,” Algebra and Logic, vol. 55, No. 5, 354–366 (2016).
Grechkoseeva M. A. and Staroletov A. M., “Unrecognizability by spectrum of finite simple orthogonal groups of dimension nine,” Sib. Èlektron. Mat. Izv., vol. 11, 921–928 (2014).
Mazurov V. D., “Recognition of finite groups by a set of orders of their elements,” Algebra and Logic, vol. 37, No. 6, 371–379 (1998).
Bray J., Holt D., and Roney-Dougal C., The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, Camb. Univ. Press, Cambridge (2009).
Britnell J. R., Cycle Index Methods for Matrix Groups over Finite Fields, PhD Thes., Univ. Oxford (2003).
Gorenstein D., Lyons R., and Solomon R., The Classification of the Finite Simple Groups. Number 3, Amer. Math. Soc., Providence (1998).
Bang A. S., “Taltheoretiske Undersøgelser,” Tidsskrift Math., vol. 4, 70–80, 130–137 (1886).
Zsigmondy K., “Zur Theorie der Potenzreste,” Monatsh. Math. Phys., vol. 3, 265–284 (1892).
Buturlakin A. A., “Spectra of finite symplectic and orthogonal groups,” Siberian Adv. Math., vol. 21, No. 3, 176–210 (2011).
Buturlakin A. A. and Grechkoseeva M. A., “The cyclic structure of maximal tori of the finite classical groups,” Algebra and Logic, vol. 46, No. 2, 73–89 (2007).
Zavarnitsine A. V., “Structure of the maximal tori in spin groups,” Sib. Math. J., vol. 56, No. 3, 425–434 (2015).
Vasilev A. V. and Vdovin E. P., “An adjacency criterion for the prime graph of a finite simple group,” Algebra and Logic, vol. 44, No. 6, 381–406 (2005).
Aschbacher M., Finite Group Theory, Camb. Univ. Press, Cambridge (1986).
Vasilev A. V., Gorshkov I. B., Grechkoseeva M. A., Kondratev A. S., and Staroletov A. M., “On recognition of finite simple groups of types Bn, Cn and 2Dn for n = 2k by spectrum,” Trudy Inst. Mat. Mekh. Ural Otdel. Ross. Akad. Nauk, vol. 15, No. 2, 58–73 (2009).
Williams J. S., “Prime graph components of finite groups,” J. Algebra, vol. 69, 487–513 (1981).
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Original Russian Text© 2018 Grechkoseeva M.A.
Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, vol. 59, no. 4, pp. 791–813, July–August, 2018; DOI: 10.17377/smzh.2018.59.405.
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Grechkoseeva, M.A. On Spectra of Almost Simple Extensions of Even-Dimensional Orthogonal Groups. Sib Math J 59, 623–640 (2018). https://doi.org/10.1134/S0037446618040055
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DOI: https://doi.org/10.1134/S0037446618040055