Let $G$ be a group and $H\le G$. If there exists a homomorphism $f:G\to H$ such that $f(h)=h$ for all $h\in H$, then is $H$ normal in $G$?

Question

Let $G$ be a group and $H$ be a subgroup of $G$. If there exists a homomorphism $f:G\to H$ such that $f(h)=h$ for all $h\in H$, then is $H$ normal?

Answer 1

Let's see. What is the definition of normal subgroup? $H$ is normal in $G$ if $g^{-1}hg\in H~\forall~h\in H, g\in G$. Since the only thing we have is $f$ there is only one reasonable path, which is to take some $h\in H,g\in G$ and to compute: $$f(g^{-1}hg)=f(g)^{-1}f(h)f(g)=f(g)^{-1}hf(g)$$ Now, since we want to study if $g^{-1}hg\in H$ it is equivalent to study if $f(g)^{-1}hf(g)=g^{-1}hg$.

From here I let you do some work. What do you think? This seems reasonable? If the answer is yes, how can you prove it? If the answer is no, find a counterexample.