Monic group morphism is an injection. I am trying to prove that a monic group morphism is an injection. Does this follow from the existence of a free group in one generator? Thank you for any thoughts on this.

Hint: If $f:G\to H$ is not injective, try to find two different subgroups of $G$ which are both mapped to the identity in $H$.
Of course, this does not work for monoids, so the idea in that case would in fact be to send the generator $1$ of $\Bbb Z$ to the two elements $a,b\in G$ which have the same image under $f$.
