It has been seen that every group of finite order can be represented as a group of permutations performed on a set of symbols whose number is equal to the order of the group. In general such a representation is possible with a smaller number of symbols. Let H be a subgroup of G, and let the operations of G be divided, in respect of H, into the sets H, S2H, S3H, ..., S_mH. If S is any operation of G, the sets SH, SS2H, SS3H, ..., SS_mH differ from the previous sets only in the sequence in which they occur. In fact, if SS_p belong to the set S_qH, then since H is a group, the set SS_pH is identical with the set S_qH. Hence, to each operation S of the group will correspond a permutation performed on the symbols of the m sets, and to the product of two operations corresponds the product of the two analogous permutations. The set of permutations, therefore, forms a group isomorphic with the given group. Moreover, the isomorphism is simple unless for one or more operations, other than identity, the sets all remain unaltered. This can only be the case for S, when every operation conjugate to S belongs to H. In this case H would contain a self-conjugate subgroup, and the isomorphism is multiple. The fact that every group of finite order can be represented, generally in several ways, as a group of permutations, gives special importance to such groups. The number of symbols involved in such a representation is called the _degree_ of the group. In accordance with the general definitions already given, a permutation-group is called transitive or intransitive according as it does or does not contain permutations changing any one of the symbols into any other. It is called imprimitive or primitive according as the symbols can or cannot be arranged in sets, such that every permutation of the group changes the symbols of any one set either among themselves or into the symbols of another set. When a group is imprimitive the number of symbols in each set must clearly be the same.