Q:

Abelian groups being isomorphic.

Suppose $G$ and $H$ are groups. For an isomorphism $f:G\to H$ to exist, is it a condition that $G$ and $H$ are either both abelian, or that neither of them are abelian?
I recall seeing such a condition somewhere ... but I can't seem to find the proposition anywhere. 

A:

For them to be isomorphic they must both be abelian or both NOT being abelian, it's easy to see why, let $G$ be abelian, then we have that
$$ab=ba$$
for any elements, our isomorphism is $\varphi$, then we get that
$$\varphi(ab)=\varphi(a)\varphi(b)=\varphi(ba)=\varphi(b)\varphi(a)$$
so they must both be abelian then if one is.

