Encyclopaedia Britannica, 11th Edition, "Groups, Theory of" to "Gwyniad" Volume 12, Slice 6

GROUPS,[1] THEORY OF. The conception of an operation to be carried out on some object or set of objects underlies all mathematical science. Thus in elementary arithmetic there are the fundamental operations of the addition and the multiplication of integers; in algebra a linear transformation is an operation which may be carried out on any set of variables; while in geometry a translation, a rotation, or a projective transformation are operations which may be carried out on any figure.

In speaking of an operation, an object or a set of objects to which it may be applied is postulated; and the operation may, and generally will, have no meaning except in regard to such a set of objects. If two operations, which can be performed on the same set of objects, are such that, when carried out in succession on any possible object, the result, whichever operation is performed first, is to produce no change in the object, then each of the operations is spoken of as a definite operation, and each of them is called the inverse of the other. Thus the operations which consist in replacing x by nx and by x/n respectively, in any rational function of x, are definite inverse operations, if n is any assigned number except zero. On the contrary, the operation of replacing x by an assigned number in any rational function of x is not, in the present sense, although it leads to a unique result, a definite operation; there is in fact no unique inverse operation corresponding to it. It is to be noticed that the question whether an operation is a definite operation or no may depend on the range of the objects on which it operates. For example, the operations of squaring and extracting the square root are definite inverse operations if the objects are restricted to be real positive numbers, but not otherwise.

If O, O', O", ... is the totality of the objects on which a definite operation S and its inverse S' may be carried out, and if the result of carrying out S on O is represented by O.S, then O.S.S', O.S'.S, and O are the same object whatever object of the set O may be. This will be represented by the equations SS' = S'S = 1. Now O.S.S' has a meaning only if O.S is an object on which S' may be performed. Hence whatever object of the set O may be, both O.S and O.S' belong to the set. Similarly O.S.S, O.S.S.S, ... are objects of the set. These will be represented by O.S^2, O.S^3, ... Suppose now that T is another definite operation with the same set of objects as S, and that T' is its inverse operation. Then O.S.T is a definite operation of the set, and therefore the result of carrying out S and then T on the set of objects is some operation U with a unique result. Represent by U' the result of carrying out T' and then S'. Then O.UU' = O.S.T.T'.S' = O.SS' = O, and O.U'U = O.T'.S'.S.T = O.T'T = O, whatever object O may be. Hence UU' = U'U = 1; and U, U' are definite inverse operations.

If S, U, V are definite operations, and if S' is the inverse of S, then