Tests for the nonsimplicity of factorable groups

L. S. Kazarin

Izvestiya. Mathematics , Tome 16 (1981) no. 2, pp. 261-278

Résumé

The following theorem is proved. Theorem. Suppose that a finite group $G$ is the product of two subgroups $A$ and $B,$ where $B$ is of odd order. Let at least one of the following conditions be satisfied: (a) $A$ is $2$-separable, and $(|A|,|B|)=1$. (b) $A$ is $2$-nilpotent with a $2$-separable derived group, $B$ is nilpotent, and $(|A|,|B|)=1$. (c) $A$ is supersolvable and $B$ is nilpotent. \noindent Then $O(A)$ lies in $O(G)$. Bibliography: 30 titles.