Am a novice, & am practicing to have hold of the subject. Also, request source with such questions as examples.
Need show all the $27$ cases for a group's associativity property, with three elements.
As know by group table of $3$ elements that there can be only 3 elements $e, b,c$, as below:
$$ \begin{array}{c|lcr} & e & a & b \\ \hline e & e & a & b \\ a & a & b & e \\ b & b & e & a \end{array}$$
For a group $G$ described by its elements, here $3$, & the operation (let, *), the associative property implies: $\alpha*(\beta*\gamma) = (\alpha*\beta)*\gamma$.
As can have for each of the $3$ elements : $\alpha, \beta, \gamma$, the values : $e, a, b$; so the possibilities are : $3*3*3 =27$.
For the group however, can have only $6$ cases as shown below:
(i) $a*(b*e) = (a*b)*e$,
(ii) $a*(e*b) = (a*e)*b$,
(iii) $b*(a*e) = (b*a)*e$,
(iv) $b*(e*a) = (b*e)*a$,
(v) $e*(a*b) = (e*a)*b$,
(vi) $e*(b*a) = (e*b)*a$,
There is no way, can proceed to get 21 more cases in associativity.
But, it is given in the book 'Abstract Algebra: A First Undergraduate Course' , by 'Hillman', on pg. #57 that there are $27$ cases to check for associativity of a group of $3$ elements, as shown below:
I can only think of $27$ cases for checking associativity when any of the $3$ elements : $\alpha, \beta, \gamma$ can be $a,b,e$ simultaneously, i.e it is possible to have a case like : $a*(a*a)$, & so on.
Below are the $27$ possible cases with that possibility, but I feel that these are invalid for the case of group, as all $3$ elements should be different:
(vii) $a*(a*a) = (a*a)*a$,
(viii) $b*(b*b) = (b*b)*b$,
(ix) $e*(e*e) = (e*e)*e$,
(x) $a*(b*a) = (a*b)*a$, --- (xi) $a*(a*b) = (a*a)*b$
(xii) $a*(e*a) = (a*e)*a$, --- (xiii) $a*(a*e) = (a*a)*e$
(xiv) $e*(b*a) = (e*b)*a$, --- (xv) $e*(a*b) = (e*a)*b$
(xvi) $e*(e*a) = (e*e)*a$, --- (xvii) $e*(a*e) = (e*a)*e$
(xviii) $b*(b*a) = (b*b)*a$, --- (xix) $b*(a*b) = (b*a)*b$,
(xx) $b*(e*a) = (b*e)*a$, --- (xxi) $b*(a*e) = (b*a)*e$,
(xxii) $a*(b*b) = (a*b)*b$,
(xxiii) $a*(e*e) = (a*e)*e$,
(xxiv) $b*(a*a) = (b*a)*a$,
(xxv) $b*(e*e) = (b*e)*e$,
(xxvi) $e*(a*a) = (e*a)*a$,
(xxvii) $e*(b*b) = (e*b)*b$,