Prove G is abelian. Let G be a group. Prove or disprove: We have $(g_1 * g_2)* g_3 = (g_3 * g_2) * g_1 \forall g_1, g_2, g_3 \in G$ if and only if G is abelian.
I am currently studying my old notes but it seems I found this intuitively easy to prove before that I did not write out a formal proof, but am unable to prove it now. Any suggestions are helpful.

If you put $g_3=1$ in the equality, you have $g_1g_2=g_2g_1$.
