Recognizability of groups $E_7(2)$ and $E_7(3)$ by prime graph
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 2, pp. 223-229 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

It is proved that the groups $E_7(2)$ and $E_7(3)$ are recognized by prime graph. As corollary, it is completed the positive solution of the Mazurov's problem that every finite group whose the prime graph has at least three connected components is either recognizable by spectrum or isomorphic to $A_6$.
Keywords: finite group, spectrum, prime graph, recognition by spectrum or by prime graph.
Mots-clés : simple group
@article{TIMM_2014_20_2_a17,
     author = {A. S. Kondrat'ev},
     title = {Recognizability of groups $E_7(2)$ and $E_7(3)$ by prime graph},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {223--229},
     year = {2014},
     volume = {20},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2014_20_2_a17/}
}
TY  - JOUR
AU  - A. S. Kondrat'ev
TI  - Recognizability of groups $E_7(2)$ and $E_7(3)$ by prime graph
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2014
SP  - 223
EP  - 229
VL  - 20
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMM_2014_20_2_a17/
LA  - ru
ID  - TIMM_2014_20_2_a17
ER  - 
%0 Journal Article
%A A. S. Kondrat'ev
%T Recognizability of groups $E_7(2)$ and $E_7(3)$ by prime graph
%J Trudy Instituta matematiki i mehaniki
%D 2014
%P 223-229
%V 20
%N 2
%U http://geodesic.mathdoc.fr/item/TIMM_2014_20_2_a17/
%G ru
%F TIMM_2014_20_2_a17
A. S. Kondrat'ev. Recognizability of groups $E_7(2)$ and $E_7(3)$ by prime graph. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 2, pp. 223-229. http://geodesic.mathdoc.fr/item/TIMM_2014_20_2_a17/

[1] Alekseeva O. A., Kondratev A. S., “Kvaziraspoznavaemost nekotorykh konechnykh prostykh grupp po mnozhestvu poryadkov elementov”, Ukr. mat. kongr. Algebra i teor. chisel, Sektsiya 1, Tez. dop., Kiev, 2001, 4

[2] Alekseeva O. A., Kondratev A. S., “O raspoznavaemosti gruppy $E_8(q)$ po mnozhestvu poryadkov elementov”, Ukr. mat. zhurn., 54:7 (2002), 1003–1008 | MR

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

[4] Alekseeva O. A., Kondratev A. S., “Raspoznavaemost po spektru grupp ${^2}D_p(3)$ dlya nechetnogo prostogo chisla $p$”, Tr. In-ta matematiki i mekhaniki UrO RAN, 14, no. 4, 2008, 3–11

[5] Vasilev A. V., “Raspoznavaemost grupp $G_2(3^n)$ po poryadkam ikh elementov”, Algebra i logika, 41:2 (2002), 130–142 | MR | Zbl

[6] Vasilev A. V., “O svyazi mezhdu stroeniem konechnoi gruppy i svoistvami ee grafa prostykh chisel”, Sib. mat. zhurn., 46:3 (2005), 511–522 | MR | Zbl

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

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

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

[10] Zavarnitsin A. V., “Raspoznavanie po mnozhestvu poryadkov elementov znakoperemennykh grupp stepeni $r+1$ i $r+2$ dlya prostogo $r$ i gruppy stepeni 16”, Algebra i logika, 39:6 (2000), 648–661 | MR | Zbl

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

[12] Kondratev A. S., “Raspoznavaemost po spektru grupp $E_8(q)$”, Tr. In-ta matematiki i mekhaniki UrO RAN, 16, no. 3, 2010, 146–149

[13] Kondratev A. S., Mazurov V. D., “Raspoznavanie znakoperemennykh grupp prostoi stepeni po poryadkam ikh elementov”, Sib. mat. zhurn., 41:2 (2000), 359–369 | MR | Zbl

[14] Mazurov V. D., “Kharakterizatsii konechnykh grupp mnozhestvami poryadkov ikh elementov”, Algebra i logika, 36:1 (1997), 37–53 | MR | Zbl

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

[16] Mazurov V. D., “Gruppy s zadannym spektrom”, Izv. Ural. gos. un-ta, 36 (2005), 119–138 | MR | Zbl

[17] Mazurov V. D., Su M. Ch., Chao Kh. P., “Raspoznavanie konechnykh prostykh grupp $L_3(2^m)$ i $U_3(2^m)$ po poryadkam ikh elementov”, Algebra i logika, 39:5 (2000), 567–585 | MR | Zbl

[18] An J. B., Shi W. J., “The characterization of finite simple groups with no elements of order six”, Commun. Algebra, 28:7 (2000), 3351–3358 | DOI | MR | Zbl

[19] Aschbacher M., Finite group theory, Cambridge Univ. Press, Cambridge, 1986, 274 pp. | MR | Zbl

[20] J. H. Conway [et. al.], Atlas of finite groups, Clarendon Press, Oxford, 1985, 252 pp. | MR | Zbl

[21] Bang A. S., “Talteoretiske undersølgelser”, Tidsskrift. Math., 5:4 (1886), 70–80, 130–137 | Zbl

[22] Brandl R., Shi W. J., “Finite groups whose element orders are consecutive integers”, J. Algebra, 143:2 (1991), 388–400 | DOI | MR | Zbl

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

[24] Brandl R., Shi W. J., “The characterization of $PSL(2,q)$ by its element orders”, J. Algebra, 163:1 (1994), 109–114 | DOI | MR | Zbl

[25] Carter R. W., “Conjugacy classes in the Weyl group”, Compositio Mathematica, 25:1 (1972), 1–59 | MR | Zbl

[26] Darafsheh M. R., Moghaddamfar A. R., “A characterization of some finite groups by their element orders”, Algebra Colloq., 7:4 (2000), 467–476 | MR | Zbl

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

[28] Gorenstein D., Lyons R., Solomon R., The classification of the finite simple groups, Number 3, Part I, Math. Surveys Monogr., 40, no. 3, Amer. Math. Soc., Providence, RI, 1998, 420 pp. | MR | Zbl

[29] Guralnik R. M., Tiep P. H., “Finite simple unisingular groups of Lie type”, J. Group Theory, 6 (2003), 271–310 | MR

[30] Li H. L., Shi W. J., “A characteristic property of some sporadic simple groups”, Chinese Ann. Math., 14A:2 (1993), 144–151 (in Chinese) | MR | Zbl

[31] Lipschutz S., Shi W. J., “Finite groups whose element orders do not exceed twenty”, Progress in Natural Sci., 10:1 (2000), 11–21 | MR

[32] Mazurov V. D., Shi W. J., “A note to the characterization of sporadic simple groups”, Algebra Colloq., 5:3 (1998), 285–288 | MR | Zbl

[33] Praeger C. E., Shi W. J., “A characterization of some alternating and symmetric groups”, Commun. Algebra, 22:5 (1994), 1507–1530 | DOI | MR | Zbl

[34] Shi W. J., “A new characterization of some projective special linear groups and the finite groups in which every element has prime order or order $2p$”, J. Southwest-China Teachers University (N.S.), 8:1 (1983), 23–28 (in Chinese)

[35] Shi W. J., “A characteristic property of $PSL_2(7)$”, J. Austral. Math. Soc. Ser. A, 36:3 (1984), 354–356 | DOI | MR | Zbl

[36] Shi W. J., “A characterization of some $PSL_2(q)$”, J. Southwest-China Teachers University (N.S.), 10:2 (1985), 25–32 (in Chinese)

[37] Shi W. J., “A characteristic property of $A_5$”, J. Southwest-China Teachers University (N.S.), 11:3 (1986), 11–14 (in Chinese)

[38] Shi W. J., “A characteristic property of $A_8$”, Acta Mathematica Sinica (N.S.), 3:1 (1987), 92–96 | DOI | MR | Zbl

[39] Shi W. J., “A characteristic property of $J_1$ and $PSL_2(2^n)$”, Adv. in Math., 16 (1987), 397–401 (in Chinese) | Zbl

[40] Shi W. J., “A characteristic property of Mathieu groups”, Chinese Ann. Math., 9A:5 (1988), 575–580 (in Chinese) | MR | Zbl

[41] Shi W. J., “A characterization of the Conway simple group $Co_2$”, J. Math. (PRC), 9 (1989), 171–172 | MR

[42] Shi W. J., “A characterization of the Higman-Sims group”, Houston J. Math., 16:4 (1990), 597–602 | MR | Zbl

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

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

[45] Shi W. J., Li H. L., “A characteristic property of $M_{12}$ and $PSU(6,2)$”, Acta Math. Sin., 32:6 (1989), 758–764 (in Chinese) | MR | Zbl

[46] Shi W. J., Yang W. Z., “A new characterization of $A_5$ and finite groups in which every nonidentity element has prime order”, J. Southwest-China Teachers College. Ser. B, 1 (1984), 36–40 (in Chinese)

[47] Suprunenko I. D., Zalesski A. E., “Fixed vectors for elements in modules for algebraic groups”, Intern. J. Algebra Comput., 17:5–6 (2007), 1249–1261 | DOI | MR | Zbl

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

[49] Zsigmondy K., “Zur Theorie der Potenzreste”, Monatsh. Math. Phys., 3:1 (1892), 265–284 | DOI | MR | Zbl