On recognizability of some finite simple orthogonal groups by spectrum

O. A. Alekseeva ; A. S. Kondrat'ev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 1, pp. 30-43

Résumé

It is proved that if $G$ is a finite group with the same set of element orders as simple group $D_p(q)$, where $p$ is a prime, $p\ge5$ and $q\in\{2,3,5\}$, then the commutator group of $G/F(G)$ is isomorphic to $D_p(q)$, the subgroup $F(G)$ is equal to 1 for $q=5$ and to $O_q(G)$ for $q\in\{2,3\}$, $F(G)\le G'$ and $|G/G'|\le2$.

Détail

Keywords: finite simple group, spectrum of a group, prime graph, recognition by spectrum
Mots-clés : orthogonal group.

Bibliographie

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

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

[3] Mazurov V. D., “Gruppy s zadannym spektrom”, Izv. Ural. gos. un-ta. Ser. Matematika i mekhanika, 2005, no. 36, 119–138, Matematika i mekhanika. Vyp. 7 | MR

[4] Kourovskaya tetrad: Nereshennye voprosy teorii grupp, 15-e izd., In-t matematiki SO RAN, Novosibirsk, 2002, 172 pp. | MR

[5] 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

[6] 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

[7] Vasilev A. V., Grechkoseeva M.A., “O raspoznavaemosti konechnykh prostykh ortogonalnykh grupp razmernosti $2^m$, $2^m+1$ i $2^m+2$”, Sib. mat. zhurn., 45:3 (2004), 510–526 | MR

[8] Alekseeva O. A., Kondratev A. S., “Kvaziraspoznavaemost po mnozhestvu poryadkov elementov grupp ${^3}D_4(q)$ i $F_4(q)$ dlya nechetnogo $q$”, Algebra i logika, 44:5 (2005), 517–539 | MR | Zbl

[9] Alekseeva O. A., “Kvaziraspoznavaemost po mnozhestvu poryadkov elementov grupp ${^3}D_4(q)$, $q$ chetno”, Algebra i logika, 45:1 (2006), 3–19 | MR | Zbl

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

[11] 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 | MR

[12] Kondrat'ev A. S., “Recognition by spectrum of the groups ${^2}D_{2^m+1}(3)$”, Science in China. Ser. A: Mathematics, 52:1 (2009), 293–300 | DOI | MR

[13] Zinoveva M. R., Shen R., Shi V., “Raspoznavanie prostykh grupp $B_p(3)$ po mnozhestvu poryadkov elementov”, Tez. dokl. Mezhdunar. algebr. konf., posvyasch. 100-letiyu so dnya rozhdeniya A. G. Kurosha, Izd-vo mat.-mekh. fak-ta MGU, M., 2008, 105–106

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

[15] Vasilev A. V., Grechkoseeva M. A., “Raspoznavanie po spektru konechnykh prostykh lineinykh grupp malykh razmernostei nad polyami kharakteristiki 2”, Algebra i logika, 47:5 (2008), 558–570 | MR

[16] Grechkoseeva M. A., “Raspoznavaemost po spektru gruppy $\Omega_{10}^+(2)$”, Sib. mat. zhurn., 44:4 (2003), 737–741 | MR | Zbl

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

[18] J. H. Conway, Curtis R. G., Norton S. P., Parker R. A., Wilson R. A., Atlas of finite groups, Clarendon Press, Oxford, 1985, 252 pp. | MR | Zbl

[19] Seminar po algebraicheskim gruppam, Mir, M., 1973, 315 pp. | MR

[20] Steinberg R., Lektsii o gruppakh Shevalle, Mir, M., 1975, 262 pp. | MR | Zbl

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

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

[23] Gerono G. C., “Note sur la résolution en nombres entiers et positifs de l'équation $x^m=y^n+1$”, Nouv. Ann. Math. (2), 9 (1870), 469–471

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

[25] Carter R. W., “Centralizers of semisimple elements in the finite classical groups”, Proc. London Math. Soc., III. Ser., 42:1 (1981), 1–41 | DOI | MR | Zbl

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

[27] Aleeva M. R., “O konechnykh prostykh gruppakh s mnozhestvom poryadkov elementov kak u gruppy Frobeniusa ili dvoinoi gruppy Frobeniusa”, Mat. zametki, 73:3 (2003), 323–339 | MR | Zbl

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

[29] Kleidman P., Liebeck M., The subgroup structure of the finite classical groups, Cambridge Univ. Press, Cambridge, 1990, 303 pp. | MR

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

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

[32] Stensholt E., “Certain embeddings among finite groups of Lie type”, J. Algebra, 53:1 (1978), 136–187 | DOI | MR | Zbl