If x1 and x2 are any pair of distinct rational numbers, and y1 and y2 any other pair, there is just one operation of the group which changes x1 and x2 into y1 and y2 respectively. For the equations y1 = ax1 + b, y1 = ax2 + b determine a and b uniquely. The group is therefore doubly transitive in respect of the set of rational numbers. If H is the subgroup that leaves unchanged a given rational number x1, and S an operation changing x1 into x2, then every operation of S^(-1)HS leaves x2 unchanged. The subgroups, each of which leaves a single rational number unchanged, therefore form a single conjugate set. The group of multiplications leaves zero unchanged; and, as has been seen, this is conjugate with the subgroup formed of all operations x' - d = a(x - d), where d is a given rational number. This subgroup leaves d unchanged. The group of multiplications is clearly generated by the operations x' = px, where for p negative unity and each prime is taken in turn. Every addition is obtained on transforming x' = x + 1 by the different operations of the group of multiplications. Hence x' = x + 1, and x' = px, (p = -1, 3, 5, 7, ...), form a set of independent generating operations of the group. It is a discontinuous group. As a second example the group of motions in three-dimensional space will be considered. The totality of motions, i.e. of space displacements which leave the distance of every pair of points unaltered, obviously constitutes a set of operations which satisfies the group definition. From the elements of kinematics it is known that every motion is either (i.) a translation which leaves no point unaltered, but changes each of a set of parallel lines into itself; or (ii.) a rotation which leaves every point of one line unaltered and changes every other point and line; or (iii.) a twist which leaves no point and only one line (its axis) unaltered, and may be regarded as a translation along, combined with a rotation round, the axis. Let S be any motion consisting of a translation l along and a rotation a round a line AB, and let T be any other motion. There is some line CD into which T changes AB; and therefore T^(-1)ST leaves CD unchanged. Moreover, T^(-1)ST clearly effects the same translation along and rotation round CD that S effects for AB. Two motions, therefore, are conjugate if and only if the amplitudes of their translation and rotation components are respectively equal. In particular, all translations of equal amplitude are conjugate, as also are all rotations of equal amplitude. Any two translations are permutable with each other, and give when combined another translation. The totality of translations constitutes, therefore, a subgroup of the general group of motions; and this subgroup is a self-conjugate subgroup, since a translation is always conjugate to a translation.