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If $H$ is a quotient of $G$, does there exist an $H$-extension of $\mathbb{Q}$ not contained...

Let $\phi\colon G\rightarrow H$ be a surjective homomorphism between finite groups. Assume that $\phi$ is not split, in other words there exists no homomorphism $\sigma\colon H\rightarrow G$ such that …
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