Q:

How to prove this binary operations

Let $(G, ∗)$ be a group and $a,b \in G$. 
Show that if $(a∗b)∗(a∗b) = (a∗a)∗(b∗b)$, then $(a∗b)=(b∗a)$.
How can ve prove this? Thanks for your help..

A:

Since it is a group, the operation $\cdot$ is associative, hence you have
$(a \cdot b)\cdot(b\cdot a)=a\cdot(b\cdot a)\cdot b$
By your assumption you know it equals
$(a \cdot a) \cdot ( b \cdot b)= a \cdot (a \cdot b) \cdot b$
Then you have
$a\cdot(b\cdot a)\cdot b = a \cdot (a \cdot b) \cdot b$
Since it is a group, there are inverse $a^{-1}$ and $b^{-1}$. You multiply the equation on the left by $a^{-1}$ and on the right by $b^{-1}$ and you get the solution.

