Tests for the nonsimplicity of factorable groups

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 $O(G)$.
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