Let $G,H$ be groups and $\phi\colon G\to H$ be a one-to-one group homomorphism. Let $a\in G$. Show that $|\phi(a)| \mid |G|$. Not sure how to attack this proof.
Do we use the First Isomorphism Theorem and Lagrange's Theorem? I am unsure if the canonical homomorphism is helpful for this proof.