Abstract
We obtain the first example of an infinite series of finite simple groups that are uniquely determined by their prime graph in the class of all finite groups. We also show that there exist almost simple groups for which the number of finite groups with the same prime graph is equal to 2.
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References
V. D. Mazurov, “Groups with prescribed orders of elements,” Izv. Ural. Gos. Univ., Mat. Mekh., 36, No. 7, 119–138 (2005).
A. Khosravi and B. Khosravi, “Quasirecognition of the simple group 2G2(q) by the prime graph,” to appear in Sib. Mat. Zh.
J. S. Williams, “Prime graph components of finite groups,” J. Alg., 69, No. 2, 487–513 (1981).
A. S. Kondratiev, “On prime graph components for finite simple groups,” Mat. Sb., 180, No. 6, 787–797 (1989).
W. Shi, “The characterization of the sporadic simple groups by their element orders,” Alg. Coll., 1, No. 2, 159–166 (1994).
A. V. Zavarnitsine, “Recognition of the simple groups L 3(q) by element orders,” J. Group Theory, 7, No. 1, 81–97 (2004).
V. D. Mazurov, M. C. Xu, and H. P. Cao, “Recognition of finite simple groups L 3(2m) and U 3(2m) by their element orders,” Algebra Logika, 39, No. 5, 567–585 (2000).
V. D. Mazurov, “Recognition of finite simple groups S 4(q) by their element orders,” Algebra Logika, 41, No. 2, 166–198 (2002).
V. D. Mazurov, “The set of orders of elements in a finite group,” Algebra Logika, 33, No. 1, 81–89 (1994).
K. Zsigmondy, “Zur Theorie der Potenzreste,” Mon. Math. Phys., 3, 265–284 (1892).
The GAP Group, GAP — Groups, Algorithms, and Programming, Vers. 4.4.7 (2006); http: //www.gap-system.org.
R. M. Guralnick and P. H. Tiep, “Finite simple unisingular groups of Lie type,” J. Group Theory, 6, No. 3, 271–310 (2003).
B. Chang and R. Ree, “The character of G 2(q),” in Symp. Math., Vol. 13, Academic Press, London (1974), pp. 395–413.
G. Hiss and J. Shamash, “3-blocks and 3-modular characters of G 2(q),” J. Alg., 131, No. 2, 371–387 (1990).
H. N. Ward, “On Ree’s series of simple groups,” Trans. Am. Math. Soc., 121, No. 1, 62–89 (1966).
P. Landrock and G. O. Michler, “Principal 2-blocks of the simple groups of Ree type,” Trans. Am. Math. Soc., 260, 83–111 (1980).
B. Huppert and N. Blackburn, Finite Groups, Vol. 3, Springer, Berlin (1982).
J. Conway, R. Curtis, S. Norton, et al., Atlas of Finite Groups, Clarendon, Oxford (1985).
M. Hagie, “The prime graph of a sporadic simple group,” Comm. Alg., 31, No. 9, 4405–4424 (2003).
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Supported by RFBR grant No. 05-01-00797, and by SB RAS Young Researchers Support grant No. 29 and Integration project No. 2006.1.2.
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Translated from Algebra i Logika, Vol. 45, No. 4, pp. 390–408, July–August, 2006.
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Zavarnitsine, A.V. Recognition of finite groups by the prime graph. Algebr Logic 45, 220–231 (2006). https://doi.org/10.1007/s10469-006-0020-9
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DOI: https://doi.org/10.1007/s10469-006-0020-9