Similarly
US = VS
implies
U = V.
Let S, T, U, ... be a set of definite operations, capable of being Definition of a group. 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.
If S′ is the inverse operation of S, a group which contains S must contain SS′, which produces no change on any possible object. This is called the identical operation, and will always be represented by I. Since SpSq = Sp+q when p and q are positive integers, and SpS′ = Sp−1 while no meaning at present has been attached to Sq when q is negative, S′ may be consistently represented by S−1. The set of operations ..., S−2, S−1, 1, S, S2, ... obviously constitute a group. Such a group is called a cyclical group.
It will be convenient, before giving some illustrations of the general group idea, to add a number of further definitions and explanations which apply to all groups alike. If from among the set of operations S, T, U, ... which constitute a group Subgroups, conjugate operations, isomorphism, &c. G, a smaller set S′, T′, U′, ... can be chosen which themselves constitute a group H, the group H is called a subgroup of G. Thus, in particular, if S is an operation of G, the cyclical group constituted by ..., S−2, S−1, 1, S, S2, ... is a subgroup of G, except in the special case when it coincides with G itself.