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 SU = SV implies S'SU = S'SV, or U = V. Similarly US = VS implies U = V. Definition of a group. Let S, T, U, ... be a set of definite operations, capable of being carried out on a common object or set of objects, and let the set contain-- (i.) the operation ST, S and T being any two operations of the set; (ii.) the inverse operation of S, S being any operation of the set; the set of operations is then called a group. The number of operations in a group may be either finite or infinite. When it is finite, the number is called the _order_ of the group, and the group is spoken of as a _group of finite order_. If the number of operations is infinite, there are three possible cases. When the group is represented by a set of geometrical operations, for the specification of an individual operation a number of measurements will be necessary. In more analytical language, each operation will be specified by the values of a set of parameters. If no one of these parameters is capable of continuous variation, the group is called a _discontinuous group_. If all the parameters are capable of continuous variation, the group is called a _continuous group_. If some of the parameters are capable of continuous variation and some are not, the group is called a _mixed group_.