Given: $|G|=n$, $H$ is a subgroup of $G$ and $|G/H|=k$, where $n$ does not divide k!.
WTP: The left action map on $G/H$ has a nontrivial kernel.
I have not put the entire problem I am trying to work on here, just the part I am stuck on. I think this should be true, but I am having trouble showing nontriviality. I know if we take an element of G/H then the kernel of the map are all elements a such that $agH=gH$, but I cannot prove that there are any $a$'s that will do this other than the identity. Can anyone help, or tell me if this is not true?