The local case in Aschbacher theorem for linear and unitary groups

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 $r=2$.