A group may be represented as isomorphic with itself by transforming all its operations by any one of them. In fact, if S_pS_q = S_r, then S^(-1)S_pS . S^(-1)S_qS = S^(-1)S_rS. An isomorphism of the group with itself, established in this way, is called an inner isomorphism. It may be regarded as an operation carried out on the symbols of the operations, being indeed a permutation performed on these symbols. The totality of these operations clearly constitutes a group isomorphic with the given group, and this group is called the group of inner isomorphisms. A group is simply or multiply isomorphic with its group of inner isomorphisms according as it does not or does contain self-conjugate operations other than identity. It may be possible to establish a correspondence between the operations of a group other than those given by the inner isomorphisms, such that if S' is the operation corresponding to S, then S'_pS'_q = S'_r is a consequence of S_pS_q = S_r. The substitution on the symbols of the operations of a group resulting from such a correspondence is called an outer isomorphism. The totality of the isomorphisms of both kinds constitutes the group of isomorphisms of the given group, and within this the group of inner isomorphisms is a self-conjugate subgroup. Every set of conjugate operations of a group is necessarily transformed into itself by an inner isomorphism, but two or more sets may be interchanged by an outer isomorphism. A subgroup of a group G, which is transformed into itself by every isomorphism of G, is called a _characteristic_ subgroup. A series of groups G, G1, G2, ..., 1, such that each is a maximum characteristic subgroup of G contained in the preceding, may be shown to have the same invariant properties as the subgroups of a composition series. A group which has no characteristic subgroup must be either a simple group or the direct product of a number of simply isomorphic simple groups. Permutation-groups.