3. (i) A sufficient condition for (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, - 00 < b < 00, 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 e D, g E G, there exists d' E D such that gd = d'g. The equality S(XI) = S(X2) implies X2 = dXI for some d e D, and hence eX2 = edx, = d'ex.. The result (i) now follows, since S is invariant under D.]