Definition : $(G,\circ)$ and $(H,\circ)$ and suppose that their underlying set is $\Omega = \{1,2,\cdots,n\}$. We say that $G$ and $H$ are isomorphic if there exists a bijection from $\phi : \Omega \mapsto \Omega$ such that $\forall a,b \in G,\phi(a\circ b) = \phi(a) \circ \phi(b) $
Question : Is this definition of group isomorphism correct? Is according to my definition $G$ is always going to isomorphic to $H$?