Testing for associativity using the multiplication table (Cayley Table) of an operation. [duplicate]

$\begingroup$

I seem to recall that there is a relatively easy method for determining the associativity of an operation by using its Cayley table. What is it?

$\endgroup$ 2

1 Answer

$\begingroup$

It is called Light's associativity test which I found on Wikipedia.

Basically,

1. Pick out the generators of the operation.
2. If $g$ is a generator define two new operations $x \circ y = (xg)y $ and $x*y=x(gy)$.
3. Form the Cayley tables of $\circ$ and $*$ for $g$.
4. If the two tables for $g$ are not identical, the original operation is NOT associative.
5. If the two tables are identical for all generators $g$, the original operation IS associative.

Notwithstanding the first comment, the link above works now, Nov 7, thanks to a kind editor.

$\endgroup$ 10