$\displaystyle K$ is a residually finite group.

By definition, for each nontrivial element in $\displaystyle K$, there exists a finite index normal subgroup $\displaystyle N_k$ in K such that $\displaystyle k \notin N_k$.

Is it possible that $\displaystyle N_k$ is charateristic in $\displaystyle K$? Why?

Definition: A subgroup $\displaystyle H$ of $\displaystyle G$ ischaracteristicin $\displaystyle G$ in case $\displaystyle f(H) \subset H$ for every isomorphism $\displaystyle f:G \longrightarrow G$.