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.
Anyone?
Consider the existence of unique inverses. What does this do to the rows and columns of the multiplication table? For example, what if $a\star b = c$ but then $a\star d = c$ too?
You'll also need an identity element, and in particular this must be a two-sided identity, meaning $e\star x = x \star e = x$. Without loss of generality we can take $e$ to be the first element by rows and columns. How should the first row and the first column be related?
As far as associativity goes, I'm not sure there's a particularly good way of describing this in terms of the group multiplication table. It just has to work. (If you can think of a good way, please let me know!)