So I have this example in my thesis that you can define an action of a group $G$ on G/H (where $H \lhd G$) like this: $\phi_g(aH)=gaH$. I thought it was simple enough that I didn't need to prove it's true, except now I tried and I can't prove it!
We say that $G$ acts on $G/H$ if there's a homomorphism $\phi: G:\to Aut G/H$, right? Well, so I need to prove two things: $\phi$ is a homomorphism from $G$ to $Aut G/H$, and $\phi_g$ is an automorphism of $G/H$. I can prove the first claim, but not the second.
If $\phi_g$ is an automorphism of $G/H$, then $$\phi_g(aHbH)=\phi_g(aH)\phi_g(bH),$$
but $\phi_g(aHbH)=\phi_g(abH)=gabH$, while $\phi_g(aH)\phi_g(bH)=gaHgbH=gagbH$, and I don't see why $gabH$ should be the same as $gagbH$. I'm clearly missing something really simple, but I really don't know what!