A question on the proof "Normal subgroup of prime index"

On Arturo's Magidin answer, it states that

Then $G/K$ is isomorphic to a subgroup of $S_p$, and so has order dividing $p!$"

Why it that true? I thought in order to have an isomorphism between $2$ groups, they must have the same order, since it's a $1-1$ and onto function.

How do you prove this?

## Best Answer

It's Lagrange's Theorem. For any subgroup $H$ of a finite group $G$, we have $$|H|\mid |G|.$$

A proof can be found in any textbook on group theory or, indeed, abstract algebra, worth its salt.