On a condition for twofold transitivity of a primitive group of permutations

The following generalization of the well-known Frobenius theorem is proved: Let $G$ be a primitive group of permutations of a set $W=X\cup Y \cup Z$. If $G$ contains a subgroup $H$ which operates trivially on $X$ and primitively on $Y$ and $Z$, and if $|X|\geqslant2$, then $G$ is doubly transitive.
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