(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, the set of operations ΣT1, ΣT2, . . ., ΣTn, or ΣH, are all distinct from each other and from the operations of H. Properties of a group which depend on the order. 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 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 Sylow’s theorem. 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.
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 p12p2, 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.
As already defined, a composite group is a group which contains one or more self-conjugate subgroups, whose orders are greater than unity. If H is a self-conjugate subgroup of G, the factor-group G/H may be either simple or composite. In the former case G can contain no self-conjugate subgroup K, Composition-series of a group. which itself contains H; for if it did K/H would be a self-conjugate subgroup of G/H. When G/H is simple, H is said to be a maximum self-conjugate subgroup of G. Suppose now that G being a given composite group, G, G1, G2, . . ., Gn, 1 is a series of subgroups of G, such that each is a maximum self-conjugate subgroup of the preceding; the last term of the series consisting of the identical operation only. Such a series is called a composition-series of G. In general it is not unique, since a group may have two or more maximum self-conjugate subgroups. A composition-series of a group, however it may be chosen, has the property that the number of terms of which it consists is always the same, while the factor-groups G/G1, G1/G2, . . ., Gn differ only in the sequence in which they occur. It should be noticed that though a group defines uniquely the set of factor-groups that occur in its composition-series, the set of factor-groups do not conversely in general define a single type of group. When the orders of all the factor-groups are primes the group is said to be soluble.
If the series of subgroups G, H, K, . . ., L, 1 is chosen so that each is the greatest self-conjugate subgroup of G contained in the previous one, the series is called a chief composition-series of G. All such series derived from a given group may be shown to consist of the same number of terms, and to give rise to the same set of factor-groups, except as regards sequence. The factor-groups of such a series will not, however, necessarily be simple groups. From any chief composition-series a composition-series may be formed by interpolating between any two terms H and K of the series for which H/K is not a simple group, a number of terms h1, h2, . . ., hr; and it may be shown that the factor-groups H/h1, h1/h2, . . ., hr/K are all simply isomorphic with each other.
A group may be represented as isomorphic with itself by transforming all its operations by any one of them. In fact, if SpSq = Sr, then S−1SpS·S−1SqS = S−1SrS. An isomorphism of the group with itself, established in this way, is called an inner isomorphism. It may be regarded as an operation Isomorphism of a group with itself. carried out on the symbols of the operations, being indeed a permutation performed on these symbols. The totality of these operations clearly constitutes a group isomorphic with the given group, and this group is called the group of inner isomorphisms. A group is simply or multiply isomorphic with its group of inner isomorphisms according as it does not or does contain self-conjugate operations other than identity. It may be possible to establish a correspondence between the operations of a group other than those given by the inner isomorphisms, such that if S′ is the operation corresponding to S, then S′pS′q = S′r is a consequence of SpSq = Sr. The substitution on the symbols of the operations of a group resulting from such a correspondence is called an outer isomorphism. The totality of the isomorphisms of both kinds constitutes the group of isomorphisms of the given group, and within this the group of inner isomorphisms is a self-conjugate subgroup. Every set of conjugate operations of a group is necessarily transformed into itself by an inner isomorphism, but two or more sets may be interchanged by an outer isomorphism.
A subgroup of a group G, which is transformed into itself by every isomorphism of G, is called a characteristic subgroup. A series of groups G, G1, G2, . . ., 1, such that each is a maximum characteristic subgroup of G contained in the preceding, may be shown to have the same invariant properties as the subgroups of a composition series. A group which has no characteristic subgroup must be either a simple
