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Let $H$ subgroup of $G$ of index prime $q$ ,if $H$ a finite $q$-residually group , prove that $G$ is finite $q$-residually group.

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Let $N$ be a normal subgroup of $H$ of index a power of the prime $q$. Since $\lvert G : N_{G}(N) \rvert \le q$ is finite, the core $$ M = \bigcap_{g \in G} N^{g} $$ of $N$ is a normal subgroup of $G$ of finite index a power of $q$. Since the intersection of all $N$'s is $1$, so is the intersection of all $M$'s.

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  • $\begingroup$ @amelz, you're welcome! $\endgroup$ – Andreas Caranti Mar 24 '15 at 10:35

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