I have this assignment from a homework that I'm pretty sure is wrong. It asks me to prove
Given a group homomorphism $\phi: G\rightarrow G'$, if $g\in G$ has order $k$ then so does $\phi(g)$.
I would think this is intuitively false with an easy counter-example: The trivial homomorphism $\phi(x)=1$ for all $x\in G$. Intuitively, you need an isomorphism for the order of an element to be preserved, right?