Abstract
The spectrum of a group is the set of its element orders. We say that the problem of recognition by spectrum is solved for a finite group if we know the number of pairwise nonisomorphic finite groups with the same spectrum as the group under study. In this article the problem of recognition by spectrum is completely solved for every finite nonabelian simple group with orders having prime divisors at most 13.
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References
Mazurov V. D., “The set of orders of elements in a finite group,” Algebra and Logic, 33, No.1, 49–56 (1994).
Mazurov V. D., “Recognition of finite groups by a set of orders of their elements,” Algebra and Logic, 37, No.6, 371–379 (1998).
Shi W., “A characteristic property of PSL 2(7),” J. Austral. Math. Soc. Ser. A, 36, No.3, 354–356 (1984).
Shi W., “A characteristic property of A 5,” J. Southwest-China Teach. Univ., 3, 11–14 (1986).
Shi W., “A characteristic property of J 1 and PSL 2(2n),” Adv. Math., 16, 397–401 (1987).
Brandl R. and Shi W., “The characterization of PSL(2,q) by its element orders, ” J. Algebra, 163, No.1, 109–114 (1994).
Mazurov V. D., “Characterizations of groups by arithmetic properties,” Algebra Colloq., 11, No.1, 129–140 (2004).
Williams J. S., “Prime graph components of finite groups,” J. Algebra, 69, No.2, 487–513 (1981).
Kondrat’ev A. S., “On prime graph components for finite simple groups,” Mat. Sb., 180, No.6, 787–797 (1989).
Kondrat’ev A. S. and Mazurov V. D., “Recognition of alternating groups of prime degree from their element orders,” Siberian. Math. J., 41, No.2, 294–302 (2000).
Mazurov V. D., “Characterizations of finite groups by sets of all orders of the elements,” Algebra and Logic, 36, No.1, 23–32 (1997).
Conway J. H., Curtis R. T., Norton S. P., Parker R. A., and Wilson R. A., Atlas of Finite Groups, Clarendon Press, Oxford (1985).
Aleeva M. R., “On finite simple groups with the set of element orders as in a Frobenius group or a double Frobenius group,” Math. Notes, 73, No.3, 299–313 (2003).
Zavarnitsin A. V., Element Orders in Coverings of the Groups L n (q) and Recognition of the Alternating Group A 16 [in Russian], NIIDMI, Novosibirsk (2000).
Jansen C., Lux K., Parker R. A., and Wilson R. A., An Atlas of Brauer Characters, Clarendon Press, Oxford (1995).
The GAP Group, GAP—Groups, Algorithms, and Programming. Version 4.4. 2004 (http:// www.gap-system.org).
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Original Russian Text Copyright © 2005 Vasil’ev A. V.
The author was supported by the Russian Foundation for Basic Research (Grants 02-01-39005; 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program “Development of the Scientific Potential of Higher School” of the Ministry for Education of the Russian Federation (Grant No. 8294), and the Program “Universities of Russia” (Grant UR.04.01.202).
Translated from Sibirski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) Matematicheski \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath}\) Zhurnal, Vol. 46, No. 2, pp. 315–324, March–April, 2005.
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Vasil’ev, A.V. On recognition of all finite nonabelian simple groups with orders having prime divisors at most 13. Sib Math J 46, 246–253 (2005). https://doi.org/10.1007/s11202-005-0024-z
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DOI: https://doi.org/10.1007/s11202-005-0024-z