Abstract
The spectrum ω(G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable by spectrum (briefly, recognizable) if H≃G for every finite group H such that ω(H)=ω(G). We give two series, infinite by dimension, of finite simple classical groups recognizable by spectrum.
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References
Mazurov V. D., “On the set of element orders in a finite group,” Algebra i Logika, 33,No. 1, 81-89 (1994).
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 Teachers 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., “Recognition of the finite simple groups S 4(q) by their element orders,” Algebra i Logika, 41,No. 2, 166-198 (2002).
Shi W. and Tang C. J., “A characterization of some orthogonal groups,” Prog. Nat. Sci., 7,No. 2, 155-162 (1997).
Mazurov V. D., Xu M. C., and Cao H. P., “Recognition of the finite simple groups L 3(2m) and U 3(2m) by their element orders,” Algebra i Logika, 39,No. 5, 567-585 (2000).
Mazurov V. D., “On recognition of finite groups by the set of element orders,” Algebra i Logika, 37,No. 6, 651-666 (1998).
Conway J. H., Curtis R. T., Norton S. P., Parker R. A., and Wilson R. A., Atlas of Finite Groups, Clarendon Press, Oxford (1985).
Alekseeva O. A. and Kondrat'ev A. S., “On recognition of the group E 8(q) by the set of element orders,” Ukrain. Mat. Zh., 54,No. 7, 998-1003 (2002).
Williams J. S., “Prime graph components of finite groups,” J. Algebra, 69,No. 2, 487-513 (1981).
Kondrat'ev A. S. and Mazurov V. D., “Recognition of alternating groups of prime degree from their element orders,” Sibirsk. Mat. Zh., 41,No. 2, 359-369 (2000).
Mazurov V. D., “Characterization of finite groups by sets of element orders,” Algebra i Logika, 36,No. 1, 37-53 (1997).
Zsigmondy K., “Zur Theorie der Potenzreste,” Monatsh. Math. Phys., 3, 265-284 (1892).
Seminar on Algebraic Groups and Related Finite Groups, Springer-Verlag, Berlin; Heidelberg; New York (1970).
Carter R. W., Simple Groups of Lie Type, John Wiley & Sons, London (1972).
Zavarnitsin A. V., Element Orders in Coverings of the Groups L n (q) and Recognition of the Alternating Group A 16 [in Russian] [Preprint, No. 48], NII Diskret. Mat. Inform., Novosibirsk (2000).
Grechkoseeva M. A., “On minimal permutation representations of classical simple groups,” Sibirsk. Mat. Zh., 44,No. 3, 560-586 (2003).
Aleeva M. R., “On finite simple groups with the set of element orders as that of the Frobenius or double Frobenius group,” Mat. Zametki, 73,No. 3, 323-339 (2003).
Aschbacher M. and Seitz G. M., “Involutions in Chevalley groups over fields of even order,” Nagoya Math. J., 63, 1-91 (1976).
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Vasil'ev, A.V., Grechkoseeva, M.A. On Recognition of the Finite Simple Orthogonal Groups of Dimension 2m, 2m+1, and 2m+2 over a Field of Characteristic 2. Siberian Mathematical Journal 45, 420–432 (2004). https://doi.org/10.1023/B:SIMJ.0000028607.23176.5f
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DOI: https://doi.org/10.1023/B:SIMJ.0000028607.23176.5f