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Suppose $S$ is a finite set with a binary operation $*$ given by a Cayley table. While the commutativity of $*$ can be determined on the basis of the symmetry of the table across the upper-left to lower-right diagonal, is there any way we can, by inspecting the table alone, decide if $*$ is associative or not?

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marked as duplicate by Arnaud D., Paul Frost, Jendrik Stelzner, José Carlos Santos, max_zorn May 24 at 19:11

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Since the table is a two by two and associativity involves three elements, it is not as easy to check from the Cayley table. You could brute force it by checking every product of three elements. There is Light's associativity test: http://en.wikipedia.org/wiki/Light%27s_associativity_test

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Wikipedia says

It is not generally possible to determine whether or not an operation is associative simply by glancing at its Cayley table. [...] However, Light's associativity test can determine associativity with less effort than brute force.

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