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.