Sur le groupe des automorphismes d'un groupoide

In some of the articles quoted in the present pater it is proved that every group can be represented as the automorphism group of a groupoid. The groupoid constructed in [1] is a semilattice, in [2] the groupoid is a commutative monoid, in [9] it is a semigroup, and in [8] it is a commutative groupoid. In particular, in [5] M. Gould proves that every finite or countable group G is isomorphic to the automorphism group of a left-cancellative groupoid (G, *). In the paper, using the faithful actions of a group on a set X, we define binary operations * in such a way that the corresponding grupoid (X, *) is, e.g., divisible from the left, left quasigroup, commutative, cyclic or has left cancellation, idempotents, left or righ identities. Of special interest is the case in which the action of G over X is regular, for, in this case, we show that, if X is finite or countable, then G is isomorphic to the automorphism group of a left cancellation grupoid (X, *). In particular we obtain another proof of Gould's theorem using the left regular representation of a group G on itself.