Factorizations of the groups of Lie type of Lie rank three over fields of 2 or 3 elements

Abstract

The following result is proved.

Let $G$ be a group of Lie type of Lie rank three over a field of 2 or 3 elements. Suppose that $G=AB$, where $A,B$ are proper non-Abelian simple subgroups of $G$. Then one of the following holds:

1) $G=L_4(2),A\cong L_3(2),B\cong A_6$ or $A_7$;

2)$G=L_4(3),A\cong L_3(3),B\cong S_4(3)$;

3) $G=S_6(2),A\cong L_4(2),B\cong L_2(8)$ or $A\cong U_4(2),B\cong L_2(8)$ or $U_3(3)$;

4) $G=U_6(2),A\cong U_5(2),B\cong S_6(2),U_4(3)$ or $M_{22}$

5) $G=U_6(3),A\cong U_5(3),B\cong S_6(3)$;

6) $G=O_7(3),A\cong L_4(3),B\cong U_3(3),G_2(3),S_6(2)$ or $A_9$ or $A\cong G_2(3),B\cong S_4(3),S_6(2)$