I have the following problem
Let $G$ and $G_1$ be finite groups such that $\gcd(|G|,|G_1|)=1$, and let $\phi:G\to G_1$ be a group homomorphism. Prove that $ker \phi=G$.
We can assume $\ker\phi$ is a subgroup of $G$.
My attempted proof is as follows:
It would be highly appreciated if someone could help to prove this as I am preparing for a group theory test. Thanks :)