I recently came across this problem:

Consider $\mathbb{Z}\_8^\*$, which are all the elements of $\mathbb{Z}\_8$ that has a multiplicative inverse. So, $\mathbb{Z}\_8^\*$ = {1, 3, 5, 7}. Now also consider $\mathbb{Z}\_{10}^\*$ = {1, 3, 7, 9}, and $\mathbb{Z}\_{12}^\*$ = {1, 5, 7, 11}. Is $\mathbb{Z}\_8^\*$ isomorphic to $\mathbb{Z}\_{10}^\*$ or $\mathbb{Z}\_{12}^\*$? Is $\mathbb{Z}\_{10}^\*$ isomorphic to $\mathbb{Z}\_{12}^\*$?

I have discovered that $\mathbb{Z}\_8^\*$ and $\mathbb{Z}\_{12}^\*$ are abelian, while $\mathbb{Z}\_{10}^\*$ is not. Also, I can see that $\mathbb{Z}\_8^\*$ and $\mathbb{Z}\_{10}^\*$ are not isomorphic. Is it safe to say that for this case, abelian groups cannot be isomorphic to non-abelian groups? If not, how should I approach this problem?? Thank you.