Let H be a subgroup of finite group G. G acts on G/H by left multiplication. This induces a homomorphism. Show that its kernel is in H

Question

Let G be a finite group and H is a subgroup of G. We have G acts on the set of left co-sets of H (G/H) by left multiplication x(gH)=xgH. This action induces a homomorphism from G to perm(G/H). Show that the kernel of this homomorphism is in H.

Can someone please explain this question? I don't know where to start. I know that the kernel of this homomorphism is the subgroup of G. So it also acts on G by the left multiplication.

Answer 1

Suppose $gg_1H=g_1H\,,\forall g_1\in G$, that is $g\in\operatorname{ker}\phi$, where $\phi$ is your homomorphism. Then , in particular, $gH=H$. So $g\in H$.