where the a’s are constants, expressing the new integrals in terms of the original ones. To each closed path described by x0 there therefore corresponds a definite linear substitution performed on the y’s. Further, if S1 and S2 are the substitutions that correspond to two closed paths L1 and L2, then to any closed path which can be continuously deformed, without crossing a singular point, into L1 followed by L2, there corresponds the substitution S1S2. Let L1, L2, ..., Lr be arbitrarily chosen closed paths starting from and returning to the same point, and each of them enclosing a single one of the (r) finite singular points of the equation. Every closed path in the plane can be formed by combinations of these r paths taken either in the positive or in the negative direction. Also a closed path which does not cut itself, and encloses all the r singular points within it, is equivalent to a path enclosing the point at infinity and no finite singular point. If S1, S2, S3, ..., Sr are the linear substitutions that correspond to these r paths, then the substitution corresponding to every possible path can be obtained by combination and repetition of these r substitutions, and they therefore generate a discontinuous group each of whose operations corresponds to a definite closed path. The group thus arrived at is called the group of the equation. For a given equation it is unique in type. In fact, the only effect of starting from another set of independent integrals is to transform every operation of the group by an arbitrary substitution, while choosing a different set of paths is equivalent to taking a new set of generating operations. The great importance of the group of the equation in connexion with the nature of its integrals cannot here be dealt with, but it may be pointed out that if all the integrals of the equation are algebraic functions, the group must be a group of finite order, since the set of quantities y1, y2 ..., yn can then only take a finite number of distinct values.
Groups of Finite Order.
We shall now pass on to groups of finite order. It is clear that here we must have to do with many properties which have no direct analogues in the theory of continuous groups or in that of discontinuous groups in general; those properties, namely, which depend on the fact that the number of distinct operations in the group is finite.
Let S1, S2, S3, ..., SN denote the operations of a group G of finite order N, S1 being the identical operation. The tableau
| S1, | S2, | S3, | ..., | SN, |
| S1S2, | S2S2, | S3S3, | ..., | SNS2, |
| S1S3, | S2S3, | S3S3, | ..., | SNS3, |
| · | · | · | · | · |
| S1SN, | S2SN, | S3SN, | ..., | 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, ..., Tn are the operations of a subgroup H of G, and if Σ is any operation of G which is not contained in H, Properties of a group which depend on the order. the set of operations ΣT1, ΣT2, ..., ΣTn, or ΣH, are all distinct from each other and from the operations of H. If the sets H and ΣH do not exhaust the operations of G, and if Σ′ is an operation not belonging to them, then the operations of the set Σ′H are distinct from each other and from those of H and ΣH. 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 HΣ 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.
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 Sylow’s theorem. 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 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 1 + 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 1 + 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.