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 S^pS^q = S^(p+q) when p and q are positive integers, and S^pS' = S^(p-1) while no meaning at present has been attached to S^q when q is negative, S' may be consistently represented by S^(-1). The set of operations ..., S^(-2), S(-1), 1, S, S^2, ... obviously constitute a group. Such a group is called a _cyclical_ group. Subgroups, conjugate operations, isomorphism, &c. 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 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, S^2, ... is a subgroup of G, except in the special case when it coincides with G itself. If S and T are any two operations of G, the two operations S and T^(-1)ST are called _conjugate_ operations, and T^(-1)ST is spoken of as the result of _transforming_ S by T. It is to be noted that since ST = T^(-1), TS, T, ST and TS are always conjugate operations in any group containing both S and T. If T transforms S into itself, that is, if S = T^(-1)ST or TS = ST, S and T are called _permutable_ operations. A group whose operations are all permutable with each other is called an _Abelian_ group. If S is transformed into itself by every operation of G, or, in other words, if it is permutable with every operation of G, it is called a _self-conjugate_ operation of G. The conception of operations being conjugate to each other is extended to subgroups. If S', T', U', ... are the operations of a subgroup H, and if R is any operation of G, then the operations R^(-1)S'R, R^(-1)T'R, R^(-1)U'R, ... belong to G, and constitute a subgroup of G. For if S'T' = U', then R^(-1)S'R.R^(-1)T'R = R^(-1)S'T'R = R^(-1)U'R. This subgroup may be identical with H. In particular, it is necessarily the same as H if R belongs to H. If it is not identical with H, it is said to be _conjugate_ to H; and it is in any case represented by the symbol R^(-1)HR. If H = R^(-1)HR, the operation R is said to be permutable with the subgroup H. (It is to be noticed that this does not imply that R is permutable with each operation of H.) If H = R^(-1)HR, when for R is taken in turn each of the operations of G, then H is called a _self-conjugate_ subgroup of G.