On the intersections of nilpotent subgroups in finite groups with socle $L_3(q)$ or $U_3(q)$

V. I. Zenkov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 70-78

Résumé

Earlier, the author described up to conjugation all pairs $(A,B)$ of nilpotent subgroups $A$ and $B$ in a finite group $G$ with socle $L_2(q)$ for which $A\cap B^g\ne 1$ for any element $g$ of $G$. A similar description was obtained later by the author for primary subgroups $A$ and $B$ of a finite group $G$ with socle $L_n(2^m)$. In this paper, we describe up to conjugation all pairs $(A,B)$ of nilpotent subgroups $A$ and $B$ of a finite group $G$ with socle $L_3(q)$ or $U_3(q)$ for which $A\cap B^g\ne 1$ for any element $g$ of $G$. The obtained results confirm in the considered cases the hypothesis that for a finite simple non-Abelian group $G$ and its nilpotent subgroup $N$ there is an element $g\in G$ such that $N\cap N^g=1$ (Problem 15.40 from “The Kourovka Notebook”).

Détail

Keywords: finite group, nilpotent subgroup, intersection of subgroups, Fitting subgroup.

Bibliographie

[1] Isaacs I.M., Finite group theory, AMS, Providence, RI, 2008, 350 pp. | MR | Zbl

[2] Zenkov V.I., “O peresechenii abelevoi i nilpotentnoi podgrupp v konechnoi gruppe. II”, Mat. zametki, 105:3 (2019), 383–394 | MR | Zbl

[3] Mazurov V.D., Zenkov V.I., “O peresechenii silovskikh podgrupp v konechnykh gruppakh”, Algebra i logika, 35:4 (1996), 424–432 | MR | Zbl

[4] Zenkov V.I., “Peresecheniya nilpotentnykh podgrupp v konechnykh gruppakh”, Fund. i prikl. matematika, 56 (1994), 1–91

[5] Unsolved problems in group theory. The Kourovka notebook, [e-resource], 19, Novosibirsk, 2018, 250 pp. URL: http://math.nsc.ru/~alglog/19tkt.pdf

[6] Kurmazov R.K., “O peresechenii sopryazhennykh nilpotentnykh podgrupp v gruppakh podstanovok”, Sib. mat. zhurn., 54:1 (1913), 98–104 | MR

[7] Zenkov V.I., “O peresecheniyakh dvukh nilpotentnykh podgrupp v konechnykh gruppakh s tsokolem $L_2(q)$”, Sib. mat. zhurn., 57:6 (2016), 1280–1290 | MR | Zbl

[8] Zenkov V.I., “O peresecheniyakh primarnykh podgrupp v nerazreshimykh konechnykh gruppakh s tsokolem, izomorfnym $L_n(2^m)$”, Sib. mat. zhurn., 59:2 (2018), 337–344 | MR | Zbl

[9] Gorenstein D., Konechnye prostye gruppy. Vvedenie v ikh klassifikatsiyu, Mir, M., 1985, 352 pp.

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

[11] Gorenstein D., Lyons R., “The local structure of finite groups of characteristic 2 type”, Mem. Amer. Math. Soc., 42, 1983, 1–731 | MR

[12] Zenkov V.I., “O peresechenii dvukh nilpotentnykh podgrupp v nebolshikh konechnykh gruppakh”, Sib. elektron. mat. izv., 13 (2016), 1099–1115 | Zbl

[13] Gorenstein D., Lyons R., Solomon R., The classification of the finite simple groups, Number 3, AMS, Providence, RI, 1998, 420 pp. | MR

[14] Aschbacher M., Seitz G., “Involutions in Chevalley groups over fields of even order”, Nagoya Math. J., 60 (1974), 1–91 | MR

[15] Yong Yang, “Regular orbits of nilpotent subgroups of solvable linear groups”, J. Algebra, 325:1 (2011), 56–69 | DOI | MR | Zbl

[16] Aschbacher M., “Overgroups of Sylow subgroups in sporadic groups”, Mem. Amer. Math. Soc., 60, no. 343, 1986, 1–235 | DOI | MR

[17] Gagen T., “K teorii konechnykh grupp”, Nekotorye voprosy teorii konechnykh grupp, Mir, M., 1979, 13–96