TITLE: If $\phi: G\rightarrow G'$ be a group onto homomorphism then show that $|G'|$ divides $|G|$
QUESTION [0 upvotes]: If $\phi: G\rightarrow G'$ be a (finite) group onto homomorphism then show that $|G'|$ divides $|G|$ 

$\phi: G\rightarrow G'$ be a group onto homomorphism then by Isomorphism theorem,
$G/Ker~ \phi \simeq G'$ and then $|G/Ker~ \phi|= |G'|$ i.e $$|Ker~ \phi|=\frac{|G|}{|G'|} $$
But $|G'|$ divides $|G|$ can be concluded only when $|Ker~ \phi|$ exist finitely. What to do?
Is any other alternative method to solve?

REPLY [1 votes]: It is slightly more clear to stick to $|G'| \ |Ker \phi|= |G|$, and not  to divide by $|Ker \phi|$. 
This clearly shows the divisibility in case $|G|$ is finite. If $G$ is not finite, I'd say the question does not make sense. But if one were to make sense of it then it would be again the equality $|G'| \ |Ker \phi|= |G|$ that is relevant.
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