# $G$ is finite $q$-residually group.

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

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.