FACTORIZATIONS OF THE GROUPS Ω7(q)

Abstract

The following result is proved:

Let $G={\mathrm{\Omega }}_7(q)$ and $q$ is odd. Suppose that $G=AB$, where $A,B$ are proper non-Abelian simple subgroups of $G$. Then one of the following holds:

(1) $q=3\text{ and }A\cong L_4(3)\text{ or }{G}_2(3),B\cong S{p}_6(2)\text{ or }{A}_9;$

(2) $q\equiv -1\text{ (mod 4) and }A\cong G_2(q),B\cong L_4(q);$

(3) $q\equiv 1\text{ (mod 4) and }A\cong G_2(q),B\cong U_4(q);$

(4) $q=3^{2n+1}>3\text{ and }A\cong {}^2{G}_2(q),B\cong L_4(q);$

(5) $q=3^{2n+1}\text{ and }A\cong U_3(q),B\cong L_4(q);$

(6) $q=3^{2n}\text{ and }A\cong L_3(q),B\cong U_4(q);$

(7) $A\cong G_2(q),B\cong PS{p}_4(q).$