Group theory related question

On Mathematics Stack Exchange I have posed the following question:

I am currently studying the manuscript Group Theory: A First Journey by Vipul Naik. It is available from the web page http://www.cmi.ac.in/~vipul/mathjourneys/ . 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?

After reading the comments and the answers made by other users, I began to understand what the author is asking. He wants the reader to think how would a multiplication matrix look like for a magma that follows the 3 requirements of a group. Maybe not look like, but what properties, patterns, etc. it would exhibit.