Geodesic
Spectra of automorphic extensions of finite simple exceptional groups of Lie type
Algebra i logika, Tome 55 (2016) no. 5, pp. 540-557.

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The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of elements of $G$. Let $S$ be a simple exceptional group of type $E_6$ or $E_7$. We describe all finite groups $G$ such that $S\le G\le\operatorname{Aut}S$ and $\omega(G)=\omega(S)$. Along with the previously obtained results, this provides a description of all finite groups $G$ such that $\omega(G)=\omega(S)$ and completes the study of the recognition-by-spectrum problem for all simple exceptional groups of Lie type.
Keywords: automorphic extension, finite simple group, order of element, recognizability by spectrum.
Mots-clés : exceptional group 
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M. A. Zvezdina. Spectra of automorphic extensions of finite simple exceptional groups of Lie type. Algebra i logika, Tome 55 (2016) no. 5, pp. 540-557. http://geodesic.mathdoc.fr/item/AL_2016_55_5_a1/

[1] A. V. Vasilev, A. M. Staroletov, “Pochti raspoznavaemost po spektru prostykh isklyuchitelnykh grupp lieva tipa”, Algebra i logika, 53:6 (2014), 669–692 | MR

[2] A. S. Kondratev, “Raspoznavaemost grupp $E_7(2)$ i $E_7(3)$ po grafu prostykh chisel”, Tr. IMM UrO RAN, 20, no. 2, 2014, 223–229 | MR

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

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

[5] D. Gorenstein, R. Lyons, R. Solomon, The classification of the finite simple groups. Part I. Chapter A: Almost simple $K$-groups, Math. Surv. Monogr., 40.3, Am. Math. Soc., Providence, RI, 1998 | MR | Zbl

[6] W. Shi, “A characterization of Suzuki's simple groups”, Proc. Am. Math. Soc., 114:3 (1992), 589–591 | DOI | MR | Zbl

[7] R. Brandl, W. Shi, “A characterization of finite simple groups with abelian Sylow 2-subgroups”, Ric. Mat., 42:1 (1993), 193–198 | MR | Zbl

[8] H. Deng, W. Shi, “The characterization of Ree groups $^2F_4(q)$ by their element orders”, J. Algebra, 217:1 (1999), 180–187 | DOI | MR | Zbl

[9] A. V. Vasilev, A. M. Staroletov, “Raspoznavaemost grupp $G_2(q)$ po spektru”, Algebra i logika, 52:1 (2013), 3–21 | MR | Zbl

[10] A. S. Kondratev, “Raspoznavaemost po spektru grupp $E_(q)$”, Tr. IMM UrO RAN, 16, no. 3, 2010, 146–149

[11] V. D. Mazurov, “Neraspoznavaemost konechnoi prostoi gruppy $^3D_4(2)$ po spektru”, Algebra i logika, 52:5 (2013), 601–605 | MR | Zbl

[12] M. A. Grechkoseeva, M. A. Zvezdina, “On spectra of automorphic extensions of finite simple groups $F_4(q)$ and $^3D_4(q)$”, J. Algebra Appl., 15:9 (2016), 1650168 | DOI | MR | Zbl

[13] A. V. Vasilev, M. A. Grechkoseeva, V. D. Mazurov, Kh. P. Chao, G. Yu. Chen, V. D. Shi, “Raspoznavanie konechnykh prostykh grupp $F_4(2m)$ po spektru”, Sib. matem. zh., 45:6 (2004), 1256–1262 | MR | Zbl

[14] A. S. Kondratev, “Kvaziraspoznavaemost po mnozhestvu poryadkov elementov grupp $E_6(q)$ i $^2E_6(q)$”, Sib. matem. zh., 48:6 (2007), 1250–1271 | MR | Zbl

[15] M. A. Grechkoseeva, “On element orders in covers of finite simple groups of Lie type”, J. Algebra Appl., 14:4 (2015), 1550056 | DOI | MR | Zbl

[16] O. A. Alekseeva, A. S. Kondratev, “Kvaziraspoznavaemost odnogo klassa konechnykh prostykh grupp po mnozhestvu poryadkov elementov”, Sib. matem. zh., 44:2 (2003), 241–255 | MR | Zbl

[17] A. V. Vasilev, M. A. Grechkoseeva, “Raspoznavaemost po spektru dlya prostykh klassicheskikh grupp v kharakteristike 2”, Sib. matem. zh., 56:6 (2015), 1264–1276 | MR | Zbl

[18] A. A. Buturlakin, “Spektry konechnykh prostykh grupp $E_6(q)$ i $^2E_6(q)$”, Algebra i logika, 52:3 (2013), 284–304 | MR | Zbl

[19] A. A. Buturlakin, “Spektry konechnykh prostykh grupp $E_7(q)$”, Sib. matem. zh., 57:5 (2016), 988–998 | DOI

[20] D. I. Deriziotis, A. P. Fakiolas, “The maximal tori in the finite Chevalley groups of type $E_6$, $E_7$ and $E_8$”, Commun. Algebra, 19:3 (1991), 889–903 | DOI | MR | Zbl

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

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

[23] A. V. Zavarnitsin, “Raspoznavanie prostykh grupp $U_3(q)$ po poryadkam elementov”, Algebra i logika, 45:2 (2006), 185–202 | MR | Zbl

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

[25] M. A. Grechkoseeva, “Raspoznavanie po spektru konechnykh prostykh lineinykh grupp nad polyami kharakteristiki 2”, Algebra i logika, 47:4 (2008), 405–427 | MR | Zbl

[26] K. Zsigmondy, “Zur Theorie der Potenzreste”, Monatsh. Math. Phys., 3 (1892), 265–284 | DOI | MR | Zbl

[27] R. W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Pure Appl. Math., A Wiley-Interscience Publ., John Wiley and Sons, Chichester–New York etc., 1985 | MR | Zbl

[28] M. S. Lucido, “Prime graph components of finite almost simple groups”, Rend. Semin. Mat. Univ. Padova, 102 (1999), 1–22 | MR | Zbl

[29] M. S. Lucido, “Addendum to ‘Prime graph components of finite almost simple groups’ ”, Rend. Semin. Mat. Univ. Padova, 107 (2002), 189–190 | MR | Zbl

[30] A. V. Vasilev, E. P. Vdovin, “Kriterii smezhnosti v grafe prostykh chisel konechnoi prostoi gruppy”, Algebra i logika, 44:6 (2005), 682–725 | MR | Zbl

[31] A. V. Vasilev, E. P. Vdovin, “Kokliki maksimalnogo razmera v grafe prostykh chisel konechnoi prostoi gruppy”, Algebra i logika, 50:4 (2011), 425–470 | MR | Zbl