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. 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^(-1)ST 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.