# An interesting question from “Group Theory: A First Journey,” (page 4, section 2.3).

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?

• For example (as I understand it), if you want your structure to be commutative, your table has to be symmetric, since you want $ab=ba$. If you want every element to be invertible, then you want "1"-s in certain places... – Ludolila Feb 6 '14 at 14:53
• Well, the identity and the inverse have nice enough descriptions in this context. For the associativity, it is a bit more tricky, since this involves more than two elements. – Tobias Kildetoft Feb 6 '14 at 14:55
• See Latin square. – lhf Feb 6 '14 at 15:20

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?