I am currently studying the manuscript Group Theory: A First Journey by Vipul Naik. It is available from the web page. In this manuscript the author proposes the following question:
Suppose the ordering of the elements in the rows and columns is the same. Then what kind of multiplication table would a magma have if it were to be a group? More generally, what are the constraints on the multiplication table corresponding to each of the properties that we can talk of for a binary operation?
(page 4, section 2.3)
No matter from what angle I have thought about this question, I can't seem to grasp what the author wants from his readers.