Exercise 5, § 2

For $2 \leq n \leq 5$, determine on a set E with $n$ elements all the laws everywhere defined admitting an identity element and under which all the elements of E are cancellable; for $n = 5$ show that there exist non-associative laws satisfying these conditions.

(N.B. Exercises 6 to 17 inclusive refer to associative laws written multiplicatively; $e$ denotes the identity when it exists, $\gamma_a$ and $\delta_a$ the translations by $a \in E$; for every subset X of E we write $\gamma_a(X) = aX, \delta_a(X) = Xa$.)