The order of any subgroup or operation of G is necessarily finite. If T1(= S1), T2, ..., T_n are the operations of a subgroup H of G, and if [Sigma] is any operation of G which is not contained in H, the set of operations [Sigma]T1, [Sigma]T2, ..., [Sigma]T_n, or [Sigma]H, are all distinct from each other and from the operations of H. If the sets H and [Sigma]H do not exhaust the operations of G, and if [Sigma]' is an operation not belonging to them, then the operations of the set [Sigma]'H are distinct from each other and from those of H and [Sigma]H. This process may be continued till the operations of G are exhausted. The order n of H must therefore be a factor of the order N of G. The ratio N/n is called the index of the subgroup H. By taking for H the cyclical subgroup generated by any operation S of G, it follows that the order of S must be a factor of the order of G. Every operation S is permutable with its own powers. Hence there must be some subgroup H of G of greatest possible order, such that every operation of H is permutable with S. Every operation of H transforms S into itself, and every operation of the set H[Sigma] transforms S into the same operation. Hence, when S is transformed by every operation of G, just N/n distinct operations arise if n is the order of H. These operations, and no others, are conjugate to S within G; they are said to form a set of conjugate operations. The number of operations in every conjugate set is therefore a factor of the order of G. In the same way it may be shown that the number of subgroups which are conjugate to a given subgroup is a factor of the order of G. An operation which is permutable with every operation of the group is called a _self-conjugate_ operation. The totality of the self-conjugate operations of a group forms a self-conjugate Abelian subgroup, each of whose operations is permutable with every operation of the group. Sylow's theorem.