Consider a mapping $f : G_1 \to G_2$ between two groups.
I was told that if one group is abelian and the other is not abelian, we won`t possible be able to find a bijective homomorphism between them, concluding that they are not isomorphic.
I was trying to understand formally ( by proof ) or intuitively why it is true .
Best Answer
Let $G_1$ abelian, suppose you can find an iso $f:G_1\rightarrow G_2$. Take $b_1,b_2\in G_2$ arbitrary elements. By surjectivity of $f$, there exist $a_1,a_2\in G_1$ such that $f(a_i)=b_i$. Now you get: $$b_1b_2=f(a_1)f(a_2)=f(a_1a_2)=f(a_2a_1)=f(a_2)f(a_1)=b_2b_1$$ which is abelianity of $G_2$. Hence if $G_2$ is supposed to be non-abelian, you can't find such an iso.
Actually you can't even find a surjective morphism