Given a group $G$ and a subgroup $H < G$, does conjugation given an action on cosets of $H$ via $g \cdot xH = gxHg^{-1}$?
If it does it seems there is an easier proof to the problem in Normal subgroup of prime index, where we show that a subgroup of index the smallest prime dividing $|G|$ is normal, by showing that $G$ must embed into $S_p$ trivially and therefore $g \cdot H = gHg^{-1} = H$ for all $g$, which implies that $H \lhd G$. What am I missing?
