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 $A \cong G_2(3), B \cong S_4(3), S_6(2)$