TITLE: Operation satysfaying b*(b*a)=a=(a*b)*b for all a and b must be commutative
QUESTION [6 upvotes]: Let $X$ be a nonempty set and $*$ operation defined on elements of $X$ such that for $a, b$ from $X$ there is $(a*b)*b=b*(b*a)=a$. Prove that operation $*$ is commutative. Exercise is taken from the "Introduction to algebra" by Kostrikin.

REPLY [9 votes]: $a=(a(ba))(ba)=(ba)((ba)a)=(ba)b$.
Thus, $ab=((ba)b)b=ba$