about the smallest normal subgroup containing some generators Assume $f:G\to K$ is a group homomorphism, and $S\subset G$ is a set of elements such that $f|_S$ is trivial. Denote by $N\leq G$ the smallest normal subgroup such that $S\subset N$. Is it obvious that $f$ is trivial on all of $N$?
Silly to give yet another answer for such a triviality, but none of the existing ones seem quite right.

By definition, $N$ is the intersection of the normal subgroups containing $S$. Since $\ker(f)$ is a normal subgroup and $S\subset\ker(f)$, this says that $N\subset\ker(f)$.
Tip: show that $S$ is a normal subgroup of $G$, so that $S=N$.

(In fact this is equivalent to the exercise, right?)