An Abelian group contains subgroups whose orders are any given factors of the order of the group. In fact, since every subgroup H of an Abelian group G and the corresponding factor groups G/H are Abelian, this result follows immediately by an induction from the case in which the order contains n prime factors to that in which it contains n + 1. For a group which is not Abelian no general law can be stated as to the existence or non-existence of a subgroup whose order is an arbitrarily assigned factor of the order of the group. In this connexion the most important general result, which is independent of any supposition as to the order of the group, is known as Sylow's theorem, which states that if p^a is the highest power of a prime p which divides the order of a group G, then G contains a single conjugate set of subgroups of order p^a, the number in the set being of the form 1 + kp. Sylow's theorem may be extended to show that if p^a' is a factor of the order of a group, the number of subgroups of order p^a' is of the form 1 + kp. If, however, p^a' is not the highest power of p which divides the order, these groups do not in general form a single conjugate set. The importance of Sylow's theorem in discussing the structure of a group of given order need hardly be insisted on. Thus, as a very simple instance, a group whose order is the product p1p2 of two primes (p1 < p2) must have a self-conjugate subgroup of order p2, since the order of the group contains no factor, other than unity, of the form 1 + kp2. The same again is true for a group of order p1^2p2, unless p1 = 2, and p2 = 3. There is one other numerical property of a group connected with its order which is quite general. If N is the order of G, and n a factor of N, the number of operations of G, whose orders are equal to or are factors of n, is a multiple of n. Composition-series of a group.