Factorizations of the groups PSU6(q)

Abstract

The followinf result is proved

Let $G=PS{U}_6(q)$ and $G=AB$, where $AB$ are proper non-Abelian simple subgroups of $G$. Then one of the following holds:

(1) $q=2$ and $A\cong M_{22},B\cong PS{U}_3(2);$

(2) $q=2$ and $A\cong PS{U}_4(3),B\cong PS{U}_3(2)$;

(3) $q=2^n>2,n\not\equiv 2$ (mod 4) and $A\cong PS{U}_3(q),B\cong G_2(q)$;

(4) $q\not\equiv -1$ (mod 5) and $A\cong PS{U}_5(q),B\cong PS{p}_6(q)$