## Characteristic subgroup

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

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

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