Geodesic
Recognition of Finite Groups by the Prime Graph
Algebra i logika, Tome 45 (2006) no. 4, pp. 390-408.

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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.
Keywords: prime graph, finite group, modular representations, recognition. 
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A. V. Zavarnitsin. Recognition of Finite Groups by the Prime Graph. Algebra i logika, Tome 45 (2006) no. 4, pp. 390-408. http://geodesic.mathdoc.fr/item/AL_2006_45_4_a1/

[1] V. D. Mazurov, “Gruppy s zadannym spektrom”, Izv. UrGU, mat. mekh., 36:7 (2005), 119–138 | MR

[2] A. Khosravi, B. Khosravi, “Kvaziraspoznavanie prostoi gruppy ${}^2G_2(q)$ po grafu prostykh chisel”, Sib. matem. zh., 48:3 (2007), 707–716 | MR

[3] J. S. Williams, “Prime graph components of finite groups”, J. Algebra, 69:2 (1981), 487–513 | DOI | MR | Zbl

[4] A. S. Kondratev, “O komponentakh grafa prostykh chisel konechnykh prostykh grupp”, Matem. sb., 180:6 (1989), 787–797 | MR

[5] W. Shi, “The characterization of the sporadic simple groups by their element orders”, Algebra Colloq., 1:2 (1994), 159–166 | MR | Zbl

[6] A. V. Zavarnitsine, “Recognition of the simple groups $L_3(q)$ by element orders”, J. Group Theory, 7:1 (2004), 81–97 | DOI | MR | Zbl

[7] V. D. Mazurov, M. Ch. Su, Ch. P. Chao, “Raspoznavanie konechnykh prostykh grupp $L_2(2^m)$ i $U_3(2^m)$ po poryadkam ikh elementov”, Algebra i logika, 39:5 (2000), 567–585 | MR | Zbl

[8] V. D. Mazurov, “Raspoznavanie konechnykh prostykh grupp $S_4(q)$ po poryadkam ikh elementov”, Algebra i logika, 41:2 (2002), 166–198 | MR | Zbl

[9] V. D. Mazurov, “O mnozhestve poryadkov elementov konechnoi gruppy”, Algebra i logika, 33:1 (1994), 81–89 | MR | Zbl

[10] K. Zsigmondy, “Zur Theorie der Potenzreste”, Monatsh. Math. Phys., 3 (1892), 256–284 | DOI | MR

[11] The GAP Group, GAP–Groups, algorithms, and programming, Vers. 4.4.7, , 2006 http://www.gap-system.org

[12] R. M. Guralnick, P. H. Tiep, “Finite simple unisingular groups of Lie type”, J. Group Theory, 6:3 (2003), 271–310 | DOI | MR | Zbl

[13] B. Chang, R. Ree, “The character of $G_2(q)$”, Gruppi abeliani, Gruppi e loro rappresent., Convegni 1972, Symp. Math., 13, Academic Press, London, 1974, 395–413 | MR

[14] G. Hiss, J. Shamash, “3-blocks and 3-modular characters of $G_2(q)$”, J. Algebra, 131:2 (1990), 371–387 | DOI | MR | Zbl

[15] H. N. Ward, “On Ree's series of simple groups”, Trans. Am. Math. Soc., 121:1 (1966), 62–89 | DOI | MR | Zbl

[16] P. Landrock, G. O. Michler, “Principal 2-blocks of the simple groups of Ree type”, Trans. Am. Math. Soc., 260 (1980), 83–111 | DOI | MR | Zbl

[17] B. Huppert, N. Blackburn, Finite groups III, Grundlehren Math. Wiss., 243, Springer-Verlag, Berlin a. o., 1982 | MR | Zbl

[18] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas offinite groups, Clarendon Press, Oxford, 1985 | MR | Zbl

[19] M. Hagie, “The prime graph of a sporadic simple group”, Commun. Algebra, 31:9 (2003), 4405–4424 | DOI | MR | Zbl