If O, O′, O″, ... is a set of objects in respect of which a group G is transitive, it may be possible to divide the set into a number of subsets, no two of which contain a common object, such that every operation of the group either interchanges the objects of a subset among themselves, or changes them all into the objects of some other subset. When this is the case the group is called imprimitive in respect of the set; otherwise the group is called primitive. A group which is doubly-transitive, in respect of a set of objects, obviously cannot be imprimitive.
The foregoing general definitions and explanations will now be illustrated by a consideration of certain particular groups. To begin with, as the operations involved are of the most familiar nature, the group of rational arithmetic may be considered. Illustrations of the group idea. The fundamental operations of elementary arithmetic consist in the addition and subtraction of integers, and multiplication and division by integers, division by zero alone omitted. Multiplication by zero is not a definite operation, and it must therefore be omitted in dealing with those operations of elementary arithmetic which form a group. The operation that results from carrying out additions, subtractions, multiplications and divisions, of and by integers a finite number of times, is represented by the relation x′ = ax + b, where a and b are rational numbers of which a is not zero, x is the object of the operation, and x′ is the result. The totality of operations of this form obviously constitutes a group.
If S and T represent respectively the operations x′ = ax + b and x′ = cx + d, then T−1ST represents x′ = ax + d − ad + bc. When a and b are given rational numbers, c and d may be chosen in an infinite number of ways as rational numbers, so that d − ad + bc shall be any assigned rational number. Hence the operations given by x′ = ax + b, where a is an assigned rational number and b is any rational number, are all conjugate; and no two such operations for which the a’s are different can be conjugate. If a is unity and b zero, S is the identical operation which is necessarily self-conjugate. If a is unity and b different from zero, the operation x′ = x + b is an addition. The totality of additions forms, therefore, a single conjugate set of operations. Moreover, the totality of additions with the identical operation, i.e. the totality of operations of the form x′ = x + b, where b may be any rational number or zero, obviously constitutes a group. The operations of this group are interchanged among themselves when transformed by any operation of the original group. It is therefore a self-conjugate subgroup of the original group.
The totality of multiplications, with the identical operation, i.e. all operations of the form x′ = ax, where a is any rational number other than zero, again obviously constitutes a group. This, however, is not a self-conjugate subgroup of the original group. In fact, if the operations x′ = ax are all transformed by x′ = cx + d, they give rise to the set x′ = ax + d(1 − a). When d is a given rational number, the set constitutes a subgroup which is conjugate to the group of multiplications. It is to be noticed that the operations of this latter subgroup may be written in the form x′ − d = a(x − d).
The totality of rational numbers, including zero, forms a set of objects which are interchanged among themselves by all operations of the group.
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−1HS 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−1ST leaves CD unchanged. Moreover, T-1ST 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.
All the points of space constitute a set of objects which are interchanged among themselves by all operations of the group of motions. So also do all the lines of space and all the planes. In respect of each of these sets the group is simply transitive. In fact, there is an infinite number of motions which change a point A to A′, but no motion can change A and B to A′ and B′ respectively unless the distance AB is equal to the distance A′B′.