Remark 5

Suppose that the group G is abelian. If G acts faithfully and transitively on a set X, then the stabilizer of any point of X is reduced to the identity (because the stabilizers of the points of X are all equal and their intersection is reduced to the identity element of G). Consequently, properties (i) through (vi) of Theorem 2 are also equivalent to the following:

(vii) The algebra L is étale, and G acts faithfully and transitively on $\mathscr{H}$.