I had the following question: A group of order 72 can't be a simple group. As it was asked and user Dietrich Burde gave links to already asked questions.
So, I got to know about : How to prove "a group $G$ of order $72$ can't be a simple group"?
But I have a question in answer of Ben West and the last question he answered was about 2 months ago. So, I am asking it as a separate question.
Why the kernel of the map from $G \to S_4$ cannot be trivial?
If it is trivial then what will be the contradiction?
If $G$ is a simple group then any homomorphism has kernel either $1$ or $G$. Normally, by construction the image of the map is non-trivial, so the kernel is $1$. Then $|G|=72$ and $|S_4|=24$, so the existence of an injective map is impossible.