The local case in Aschbacher theorem for linear and unitary groups

A. A. Galt ; D. O. Revin

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 1207-1218.

Résumé

Our main result completes the investigation began in [Siberian Mathematical Journal, V. 55, №2, 2014, 239–245] for linear and unitary groups. We consider the subgroups $H$ in a linear or a unitary group $G$ over a finite field such that $O_r(H)\not\nleqslant Z(G)$ for some prime $r$. We obtain a refinement of the well-known Aschbacher theorem on subgroups of classical groups for this case. More precisely, we prove that if $G=\mathrm{GL}_n^\eta(q)$, $\eta\in\{+,-\}$, $H\leqslant G$, $O_r(H)\nleqslant Z(G)$ for some prime $r$ then one of the following cases holds: $H$ is contained in some element of Aschbacher classes $\mathcal{C}_1(G)$ — $\mathcal{C}_4(G)$; $n=r^\gamma$ for a positive integer $\gamma$, $q\equiv\eta\pmod r$, $H$ is contained in the normalizer $N$ of an $r$-subgroup of symplectic type of $G$, $O_r(H)\leqslant O_r(N)$, and one of the following statements holds: $r=2$, $q\equiv-\eta\pmod4$ $N=(\mathbb{Z}_{q-\eta}\circ2_\delta^{1+2\gamma}).\mathrm{O}^\delta_{2\gamma}(2)$, $\delta\in\{+,-\}$; $N=(\mathbb{Z}_{q-\eta}\circ r^{1+2\gamma}).\mathrm{Sp}_{2\gamma}(r)$. Moreover, either $N\in\mathcal{C}_6(G)$ or $N$ is contained as a subgroup in some element of $\mathcal{C}_5(G)\cup\mathcal{C}_8(G)$. In [Siberian Mathematical Journal, V. 55, №2, 2014, 239–245] the case $r\ne 2$ was considered. Now we prove the above result for $r=2$.

Détail

Keywords: linear groups, unitary groups, Aschbacher classes, radical $2$-subgroups.

Bibliographie

[1] M. Aschbacher, “On the maximal subgroups of the finite classical groups”, Invent. math., 76 (1984), 469–514 | DOI | MR | Zbl

[2] A. Galt, W. Guo, E. M. Averkin, D. O. Revin, “On the Local Case in the Aschbacher Theorem for Linear and Unitary Groups”, Siberian Mathematical Journal, 55:2 (2014), 239–245 | DOI | MR | Zbl

[3] P. Kleidman, M. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Ser., 129, Cambridge University Press, Cambridge, 1990 | MR | Zbl

[4] J. N. Bray, D. F. Holt, C. M. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, London Math. Soc. Lecture Note Ser., 407, Cambridge University Press, Cambridge, 2013 | MR | Zbl

[5] J. L. Alperin, P. Fong, “Weights for Symmetric and General Linear Groups”, Journal of Algebra, 131 (1990), 2–22 | DOI | MR | Zbl

[6] J. An, “Weights for classical groups”, Transactions of the AMS, 342:1 (1994), 1–42 | DOI | MR | Zbl

[7] J. An, “$2$-Weights for classical groups”, J. reine angew. Math., 439 (1993), 159–204 | MR | Zbl

[8] J. An, “$2$-Weights for general linear groups”, Journal of Algebra, 149 (1992), 500–527 | DOI | MR | Zbl

[9] J. An, “$2$-Weights for unitary groups”, Transactions of the AMS, 339:1 (1993), 251–278 | MR | Zbl

[10] D. O. Revin, “The $D_{\pi}$-property in finite simple groups”, Algebra and Logic, 47:3 (2008), 210–227 | DOI | MR | Zbl

[11] D. O. Revin, “The $D_{\pi}$-property of finite groups in case $2\notin{\pi}$”, Proceedings of the Institute of Mathematics and Mechanics, 13, no. 1, 2007, 166–182 | Zbl

[12] D. O. Revin, “The $D_{\pi}$-property of linear and unitary groups”, Siberian Mathematical Journal, 49:2 (2008), 353–361 | DOI | MR | Zbl

[13] D. O. Revin, “Superlocals in Symmetric and Alternating Groups”, Algebra and Logic, 42:3 (2003), 192–206 | DOI | MR | Zbl

[14] A. Borel, J. Tits, “Eléments unipotents et sousgroupes paraboliques de groupes réductifs, I”, Invent. math., 12:2 (1971), 95–104 | DOI | MR | Zbl