Let $\phi : G \rightarrow G'$ be a group homomorphism. Show that if $|G|$ is finite, then $|\phi[G]|$ is finite and is a divisor of $|G|$.
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HINT: If $A$ is any finite set, and $f:A\to B$ is any surjection, then $B$ is finite, so the finiteness of $|\varphi[G]|$ is immediate. For the rest, use the fact that $\varphi[G]\cong G/\ker\varphi$, and $\ker\varphi$ is a subgroup of $G$.