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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?

Shaun
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Valera Rozuvan
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  • 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 06 '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 06 '14 at 14:55
  • See Latin square. – lhf Feb 06 '14 at 15:20

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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!)

Alexander Gruber
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    About testing associativity, see http://math.stackexchange.com/questions/511682/how-can-we-determine-associativity-of-a-binary-structure-from-its-cayley-table. – lhf Feb 07 '14 at 09:59