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I don't know latex commands to create the cayley table (if possible, please let me know), so consider the set $S= \{a,b,c\}$ with the binary operation $*$ defined as follows:

$a*a=a, a*b= b= b*a, a*c= c= c*a$

$b*b= a, b*c= c, c*b= b, c*c= a$

Mentally I calculated, if associativity holds, I started with the non-commutative results to get $b*c= c$ and $c*b= b$ like, $b*(b*c)= b*c= c= a*c= (b*b)*c$.

But is there any general method, rather than showing that it holds or not, verifying each and every possible combinations to show the associativity holds or doesn't holds for the whole set.

Alternatively, is there any trick to easily locate one such combination for which the associativity doesn't holds so that I can conclude Associative property doesn't works so its not even a semi-group.

Subham Karmakar
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    You may start your "research" at Light's associativity test. However, it says there that it doesn't improve the worst case runtime of naive method $\mathcal{O}(n^3)$. – Poypoyan Jun 28 '21 at 15:49
  • Since $a$ is the identity, you can easily confirm associativity for any expression involving $a$. That leaves only $8$ left to check. (Symmetry also helps reduce to basically $4$). – arbashn Jun 28 '21 at 21:08

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