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I don't see how can one check associative property in a Cayley table. For contrast, one can find identity element, inverses and commutativity. But how you check associativity?

I understand that associativity $(x \cdot y) \cdot z = x \cdot (y \cdot z)$ gives exactly one output when you take ordered triple $(x, y, z)$ and apply "$\cdot$" but I can't see how it relates to Cayley tables.

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  • It's not clear what you are asking. You can, of course, take any three elements $x$, $y$, $z$, and use the Cayley table to compute the left and right hand sides of the equation, and check whether the results are the same. What more do you want beyond that? – Lee Mosher Nov 26 '21 at 02:44
  • See Light's associativity test. – kabenyuk Nov 26 '21 at 09:33

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