Supposed theorem (from online notes): If $(G,\*)$ is a finite group and $H\subset G$, $H$ is non-empty and $H$ is closed under $\*$ then $(H,\*)$ is a group. 

The proof given is a real mess, but, after many dead ends it seems to me that the inverse of every element of H should also be in $H$. Is that true? 

I have $x\in H$, $e\in H$ and $x\*x^{-1}=e$ where $e$ is the identity element in $G$. I think I need to have $x^{-1} \in H$.