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 iff $$\left({H, *}\right)$$ is abelian.

Proof
An isomorphism is an epimorphism, so an isomorphism preserves commutativity.

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

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 $$G$$.