Isomorphism of Abelian Groups

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Theorem

Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group isomorphism.


Then $\struct {G, \circ}$ is abelian if and only if $\struct {H, *}$ is abelian.


Proof

We have that Isomorphism Preserves Commutativity.

Thus:

$\forall x, y \in G: x \circ y = y \circ x \implies \map \phi x * \map \phi y = \map \phi y * \map \phi x$


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 $G$.

$\blacksquare$


Sources