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.

Injections are the monic of the category $\mathsf{Sets}$. You then want to show that the forgetful functor $U \colon \mathsf{Grps} \to \mathsf{Sets}$ from groups to sets preserves monomorphisms.
But $U$ admits a left adjoint (namely the free group functor) and so preserves (small) limits. If you can show that the monic condition can be expressed as a limit, then you are done. I let you show that an arrow is monic if and only if [some diagram : find it !] is a limiting cone.
