A set A is endowed with a binary operation ∗ satisfying the following four conditions:
(1) If a,b,c are elements of A, then a∗(b∗c)=(a∗b)∗c ,
(2) If a,b,c are elements of A such that a∗c=b∗c, then a=b ,
(3) There exists an element e of A such that a∗e=a for all a in A, and
(4) If a and b are distinct elements of A−{e}, then a3∗b=b3∗a2, where xk=x∗xk−1 for all integers k≥2 and all x in A.
Determine the largest cardinality A may have.proposed by Bojan Basic, Serbia algebraBinary operationsetcardinality