Factorizations of the groups \(G_ 2(q)\)
Let \(G_ 2(q)\) be the (simple) group of Lie type \((G_ 2)\) over the finite field GF(q). The following result is proved. Theorem. Let \(G=G_ 2(q)\) and \(G=AB\) where A, B are proper non-Abelian simple subgroups of G. Then one of the following holds: (1) \(q=4\) and \(A\cong HJ\), \(B\cong U_ 3(4)\); (2) \(q=3^ n\) and \(A\cong L_ 3(q)\), \(B\cong U_ 3(q)\); (3) \(q=3^{2n+1}>3\) and \(A\cong L_ 3(q)\), \(B\cong^ 2G_ 2(q)\). The factorizations (1)-(3) exist. In the above, HJ is the Hall-Janko simple group, \(L_ 3(q)\) and \(U_ 3(q)\) denote respectively \(PSL_ 3(q)\) and \(PSU_ 3(q^ 2)\), and \({}^ 2G_ 2(q)\) is the Ree group of characteristic 3. This result has been proved independently U. Preiser.
scientific article; zbMATH DE number 3865550