FACTORIZATIONS OF THE GROUPS $\Omega_{7}(q)$

Abstract

The following result is proved:

Let $G=\Omega_{7}(q)$ 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) $q = 3 \textrm{ and } A \cong L_{4}(3) \textrm{ or } G_{2}(3), B \cong Sp_{6}(2) \textrm{ or } A_{9} ;$

(2) $q \equiv -1 \textrm{ (mod 4) and } A \cong G_{2}(q), B \cong L_{4}(q); $

(3) $q \equiv 1 \textrm{ (mod 4) and } A \cong G_{2}(q), B \cong U_{4}(q); $

(4) $q = 3^{2n+1} > 3 \textrm{ and } A \cong \text{ }^{2}G_{2}(q), B \cong L_{4}(q); $

(5) $q = 3^{2n+1} \textrm{ and } A \cong U_{3}(q), B \cong L_{4}(q); $

(6) $q = 3^{2n} \textrm{ and } A \cong L_{3}(q), B \cong U_{4}(q); $

(7) $A \cong G_{2}(q), B \cong PSp_{4}(q). $