Recognizability by spectrum of simple groups $C_p(3)$

M. R. Zinov'eva

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 3, pp. 88-95

Résumé

Let $G$ be a simple group, and let $\omega(G)$ be the set of orders of its elements. It is proved that, if $\omega(G)=\omega(C_p(3))$, where $p$ is an odd prime, then $G\cong C_p(3)$.

Détail

Keywords: finite simple group, spectrum of a group, prime graph, recognition by spectrum, symplectic group.

Bibliographie

[1] Mazurov V. D., “Gruppy s zadannym spektrom”, Izv. Ural. gos. un-ta. Matematika i mekhanika, 36:7 (2005), 119–138 | MR | Zbl

[2] Alekseeva O. A., Kondratev A. S., “O raspoznavaemosti gruppy $E_8(q)$ po mnozhestvu poryadkov elementov”, Ukr. mat. zhurn., 54:7 (2002), 998–1003 | MR | Zbl

[3] Vasilev A. V., “O svyazi mezhdu stroeniem konechnoi gruppy i svoistvami ee grafa prostykh chisel”, Sib. mat. zhurn., 46:2 (2005), 315–324 | MR | Zbl

[4] Vasilev A. V., Gorshkov I. B., “O raspoznavanii konechnykh prostykh grupp so svyaznym grafom prostykh chisel”, Sib. mat. zhurn., 50:2 (2009), 292–299 | MR

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

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

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

[8] 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

[9] 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

[10] Conway J. H. [et al.], Atlas of finite groups, Clarendon Press, Oxford, 1985, 250 pp. | MR | Zbl

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

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

[13] Zinoveva M. R., Shen R., Shi V., “Raspoznavanie prostykh grupp $B_p(3)$ po mnozhestvu poryadkov elementov”, Sib. mat. zhurn., 51:2 (2010), 303–315 | MR

[14] Kleidman P., Liebeck M., The subgroup structure of the finite classical groups, Cambridge Univ. Press, Cambridge, 1990 | MR | Zbl

[15] Vasilev A. V., Vdovin E. P., Kokliki maksimalnogo razmera v grafe prostykh chisel konechnoi prostoi gruppy, Preprint, In-t matematiki SO RAN, Novosibirsk, 2009, 34 pp.

[16] Lucido M. S., “Prime graph components in finite almost simple groups”, Rend. Sem. Mat. Univ. Padova, 102 (1999), 1–22 ; “Addendum to Prime graph components in finite almost simple groups”, Rend. Sem. Mat. Univ. Padova, 107 (2002), 189–190 | MR | Zbl | MR | Zbl

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

[18] ShiW. J., “Pure quantitative characterization of finite simple groups”, Front. Math. China, 2:1 (2007), 123–125 | DOI | MR