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Let N normal subgroup in $G$ of prime index $q$ we have that $N$ is residually a finite $p$-group for all primes $p$ and Hence, in particular $N$ is residually a finite $q$-group from which we get $G$ is is residually a finite $q$-group this mean that $G$ is residually nilpotent . i could'nt enderestand why?

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$\;G\;$ res. nilpotent means that for any $\;1\neq g\in G\;$ there exists $\;N_g\lhd G\;$ s.t. $\;G/N_g\;$ nilpotent and $\;g\notin N_g\;$ .

Let $\;1\neq g\in G\;$ . As $\;G\;$ is res. a finite $\;q$-group, there exists $\;N_q\lhd G\;$ s.t. $\;G/N_q\;$ is a finite $\;q$-group and $\;g\notin N_g\;$ . But finite $\;p$-groups are nilpotent and thus $\;G/N_g\;$ is nilpotent.

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