Suppose $H$ is a subgroup of a group $G$ and $aH$ is a left coset. Prove that there exists some $K$ (a subgroup of $G$) , which $aH$ is equal to $Ka$.

Question

Suppose $H$ is a subgroup of a group $G$ and $aH$ is a left coset. Prove that there exists some $K$ (a subgroup of $G$) , which $aH$ is equal to $Ka$.

I've tried to show this statement,but I cant find exactly what is $K$.

Answer 1

$$aH=Ka\iff aHa^{-1}=K$$ Aldo, note that, by a homomorphism ($x\longmapsto axa^{-1}$ is an automorphism of $G$), the image of a subgroup of $G$ is a subgroup.