I have recently taken a test and this question gave me a problem and left me confused and unsure how to answer. I did answer it although I'm pretty sure I didn't get it right. So I'm asking for help in case I see this same question on the final exam. Thanks!
Let $\varphi : G \to K$ be an epimorphism. Prove that $|K|$ divides $|G|$.
Any help is appreciated.
If you're working with ( I guess finite , since you talk about divisibility) groups, you have that (by one of the isomorphism theorems, usually the so-cold* 1st isomorphism theorem) $G/Ker(f)$~ $Im(f)=K$, then: $$|G/Ker(f)|=|G|/|Ker(f)|=|K| $$
*specially in winter.