Isomorphism of Abelian Groups

Theorem
Let $\phi: \left({G, \circ}\right) \to \left({H, *}\right)$ be a group isomorphism.

Then $\left({G, \circ}\right)$ is abelian $\left({H, *}\right)$ is abelian.

Proof
We have that an isomorphism preserves commutativity.

Thus:
 * $\forall x, y \in G: x \circ y = y \circ x \implies \phi \left({x}\right) * \phi \left({y}\right) = \phi \left({y}\right) * \phi \left({x}\right)$

Thus if $G$ is abelian, so is $H$.

As $\phi^{-1}: H \to G$ is also an isomorphism, it is clear that if $H$ is abelian, then so is $G$.