S3SN S2SN S1SN,. . . , SNSN,
when in it each compound symbol SpSQ is replaced by the single symbol Sr that is equivalent to it, is called the multiplication table of the group. It•indicates directly the result of multiplying together in an assigned sequence any number of operations of the group. In each line (and in each column) of the tableau every operation of the group occurs just once. If the letters in the tableau are regarded as mere symbols, the operation of replacing each symbol in the first line by the symbol which stands under it in the pth line is a permutation performed on the set of N symbols. Thus to the N lines of the tableau there corresponds a set of N permutations performed on the N symbols, which includes the identical permutation that leaves each unchanged. Moreover, if SpSQ=Sr,then the result of carrying out in succession the permutations which correspond to the pth and qth lines gives the permutation which corresponds to the rth line. Hence the set of permutations constitutes a group which is simply isomorphic with the given group.
Every group of finite order N can therefore be represented in concrete form as a transitive group of permutations on N symbols.
The order of any subgroup or operation of G is necessarily finite. If T1(=S1), T2, ..., T„ are the operations of a subgroup H of G,
and if E is any operation of G which is not contained in H, P r o p e r t i e s the set of operations ET1, T 2 IT,,, or EH, are all
group distinct from each other and from the operations of H. which If the sets H and EH do not exhaust the operations of G, depend on and if E' is an operation not belonging to them, then the the order. operations of the set E'H are distinct from each other and from those of H and 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 HE 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 conjugat ,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 selfconjugate operation. The totality of the selfconjugate operations of a group forms a selfconjugate Abelian subgroup, each of whose operations is permutable with every operation of the group.
An Abelian group contains subgroups whose orders are any given factors of the order of the group. In fact, since every subgroup Hof 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 n1 i. Fora group which is not Abelian no general 's law can be stated as to the existence or nonexistence of a Sylow
subgroup whose order is an arbitrarily assigned factor theorem. 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 pa 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 pa, the number in the set being of the form i +kp. Sylow's theorem may be extended to show that if pa' is a factor of the order of a group, the number of subgroups of order pa' is of the form i +kp. If, however, pa' 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 piP2 of two primes (pi 

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