Factorizations of finite groups
The main result of the paper under review is the following Theorem. Let G be a finite group which contains simple subgroups A and B with \(G=AB\). Assume that a Sylow 2-subgroup of A has rank 2 and a Sylow 2-subgroup of B is elementary Abelian. Then one of the following holds: (1) \(G=A\) or \(G=B\); (2) \(G\cong A\times B;\) (3) \(G\cong A_ 6\) and \(A\cong B\cong A_ 5;\) (4) \(G\cong A_ 7,\quad A\cong L_ 2(7)\) and \(B\cong A_ 5;\) (5) \(G\cong A_ 9,\quad A\cong A_ 6\) or \(A_ 7\), \(B\cong L_ 2(8);\) (6) \(G\cong M_{12},\quad A\cong M_{11},\quad B\cong A_ 5\) or \(L_ 2(11);\) (7) \(G\cong G_ 2(3^ m),\quad A\cong L_ 3(3^ m)\) and \(B\cong^ 2G_ 2(3^ m)\) for odd \(m\neq 1.\) The proof doesn't use the classification of finite simple groups. Remark: \textit{K. B. Tchakerian} and \textit{Ts. R. Gentchev} [Arch. Math. 41, 385-389 (1983; Zbl 0506.20011)] proved similar results in the case \(| A|_ 2\leq 8\) and \(| B|_ 2\leq 16\).
scientific article; zbMATH DE number 3851359