Group theory Definition normalizes Let $G$ be a group and consider $x\in G$ and $H\leq G$. What does it mean? "$x$ normalizes $H$".
Let $A\subseteq G$. What does it mean? "$A$ inverts $x$".

By definition
$$x\;\;\text{normalizes}\;\;H\iff H^x:=x^{-1}Hx=H\iff$$
$$\iff\;\forall\,h\in H\;\;\exists\,h'\in H\;\;s.t.\;\; x^{-1}hx=h'$$
