(i) A sufficient condition for (6.8) to hold is that D is a normal subgroup of G.

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

b, a = 0, −∞

[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.]

Section 6.3