On Finite Simple Groups with the Set of Element Orders as in a Frobenius Group or a Double Frobenius Group

M. R. Aleeva

Matematičeskie zametki, Tome 73 (2003) no. 3, pp. 323-339.

Résumé

It is proved that a finite simple group with the set of element orders as in a Frobenius group (a double Frobenius group, respectively) is isomorphic to $L_3(3)$ or $U3_(3)$ (to $U_3(3)$ or $S_4(3)$, respectively).

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”, Matem. sb., 180:6 (1989), 787–797 | Zbl

[3] Kondratev A. S., Mazurov V. D., “Raspoznavanie znakoperemennykh grupp prostoi stepeni po poryadkam ikh elementov”, Sib. matem. zh., 41:2 (2000), 359–370 | MR

[4] Aschbacher M., Finite Group Theory, Cambridge Univ. Press, Cambridge, 1986 | Zbl

[5] Conway J. H., Curtis R. T., Norton S. P., Parker R. A., Wilson R. A., An Atlas of Finite Groups, Clarendon Press, Oxford, 1985 | Zbl

[6] Thompson J. G., “Normal $p$-complements for finite groups”, Math. Z., 72:2 (1960), 332–354 | MR | Zbl

[7] Zassenhaus H., “Kennzeichnung endlicher linearean Gruppen als Permutationsgruppen”, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 17–40 | DOI

[8] Zassenhaus H., “Über endliche Fastkörper”, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 187–220 | DOI

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

[10] Mazurov V. D., Su M. Ch., Chao Ch. 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

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

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

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

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

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

[16] 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

[17] Aleeva M. R., “O kompozitsionnykh faktorakh konechnykh grupp s mnozhestvom poryadkov elementov kak u gruppy $U_3(q)$”, Sib. matem. zh., 43:2 (2002), 249–268 | MR

[18] Belonogov V. A., “Malye vzaimodeistviya v gruppakh $SL_3(q)$, $SU_3(q)$, $PSL_3(q)$ i $PSU_3(q)$”, Tr. IMM UrO RAN, 5 (1998), 3–27 | Zbl

[19] Zsigmondy H., “Zur Theorie der Potenzreste”, Monatsh. Math. Phys., 3 (1892), 265–284 | DOI

[20] Seminar po algebraicheskim gruppam, Mir, M., 1975

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

[22] Seitz G. M., “On the subgroup structure of classical groups”, Commun. Algebra, 10:8 (1982), 875–885 | DOI | MR | Zbl

[23] Deriziotis D. I., Conjugacy classes and centralizers of semisimple elements in finite groups of Lie type, Vorlesungen Fachb. Math. Univ. Essen., 11, 1984 | Zbl

[24] Srinivasan B., “The characters of the finite symplectic group $Sp(4,q)$”, Trans. Amer. Math. Soc., 131:2 (1968), 488–525 | DOI | MR | Zbl