(i) A sufficient condition for (6.9) to hold is that D is a normal subgroup of G.

(ii) If G is the group of transformations x = ax +

b, a = 0, −∞ < b < ∞, then the subgroup of translations x = x + b is normal but the subgroup x = ax is not.

[The defining property of a normal subgroup is that given d ∈ D, g ∈ G, there exists d ∈ D such that gd = d g. The equality s(x1) = s(x2) implies x2 = dx1 for some d ∈ D, and hence ex2 = edx1 = d ex1. The result (i) now follows, since s is invariant under D.]