Part 5

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.

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, 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, ..., G_n, 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, ..., G_n 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, ..., h_r; and it may be shown that the factor-groups H/h1, h1/h2, ..., h_r/K are all simply isomorphic with each other.

Isomorphism of a group with itself.

A group may be represented as isomorphic with itself by transforming all its operations by any one of them. In fact, if S_pS_q = S_r, then S^(-1)S_pS . S^(-1)S_qS = S^(-1)S_rS. An isomorphism of the group with itself, established in this way, is called an inner isomorphism. It may be regarded as an operation 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 S_pS_q = S_r. 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 group or the direct product of a number of simply isomorphic simple groups.

Permutation-groups.

It has been seen that every group of finite order can be represented as a group of permutations performed on a set of symbols whose number is equal to the order of the group. In general such a representation is possible with a smaller number of symbols. Let H be a subgroup of G, and let the operations of G be divided, in respect of H, into the sets H, S2H, S3H, ..., S_mH. If S is any operation of G, the sets SH, SS2H, SS3H, ..., SS_mH differ from the previous sets only in the sequence in which they occur. In fact, if SS_p belong to the set S_qH, then since H is a group, the set SS_pH is identical with the set S_qH. Hence, to each operation S of the group will correspond a permutation performed on the symbols of the m sets, and to the product of two operations corresponds the product of the two analogous permutations. The set of permutations, therefore, forms a group isomorphic with the given group. Moreover, the isomorphism is simple unless for one or more operations, other than identity, the sets all remain unaltered. This can only be the case for S, when every operation conjugate to S belongs to H. In this case H would contain a self-conjugate subgroup, and the isomorphism is multiple.

The fact that every group of finite order can be represented, generally in several ways, as a group of permutations, gives special importance to such groups. The number of symbols involved in such a representation is called the _degree_ of the group. In accordance with the general definitions already given, a permutation-group is called transitive or intransitive according as it does or does not contain permutations changing any one of the symbols into any other. It is called imprimitive or primitive according as the symbols can or cannot be arranged in sets, such that every permutation of the group changes the symbols of any one set either among themselves or into the symbols of another set. When a group is imprimitive the number of symbols in each set must clearly be the same.

The total number of permutations that can be performed on n symbols is n!, and these necessarily constitute a group. It is known as the _symmetric_ group of degree n, the only rational functions of the symbols which are unaltered by all possible permutations being the symmetric functions. When any permutation is carried out on the product of the n(n - 1)/2, differences of the n symbols, it must either remain unaltered or its sign must be changed. Those permutations which leave the product unaltered constitute a group of order n!/2, which is called the _alternating_ group of degree n; it is a self-conjugate subgroup of the symmetric group. Except when n = 4 the alternating group is a simple group. A group of degree n, which is not contained in the alternating group, must necessarily have a self-conjugate subgroup of index 2, consisting of those of its permutations which belong to the alternating group.

Groups of linear substitutions.

Among the various concrete forms in which a group of finite order can be presented the most important is that of a group of linear substitutions. Such groups have already been referred to in connexion with discontinuous groups. Here the number of distinct substitutions is necessarily finite; and to each operation S of a group G of finite order there will correspond a linear substitution s, viz.

__j=m x_i = \ s_(ij)x_j(i, j = 1, 2, ..., m), /__j=1

on a set of m variables, such that if ST = U, then st = u. The linear substitutions s, t, u, ... then constitute a group g with which G is isomorphic; and whether the isomorphism is simple or multiple g is said to give a "representation" of G as a group of linear substitutions. If all the substitutions of g are transformed by the same substitution on the m variables, the (in general) new group of linear substitutions so constituted is said to be "equivalent" with g as a representation of G; and two representations are called "non-equivalent," or "distinct," when one is not capable of being transformed into the other.

A group of linear substitutions on m variables is said to be "reducible" when it is possible to choose m'(< m) linear functions of the variables which are transformed among themselves by every substitution of the group. When this cannot be done the group is called "irreducible." It can be shown that a group of linear substitutions, of finite order, is always either irreducible, or such that the variables, when suitably chosen, may be divided into sets, each set being irreducibly transformed among themselves. This being so, it is clear that when the irreducible representations of a group of finite order are known, all representations may be built up.

It has been seen at the beginning of this section that every group of finite order N can be presented as a group of permutations (i.e. linear substitutions in a limited sense) on N symbols. This group is obviously reducible; in fact, the sum of the symbols remain unaltered by every substitution of the group. The fundamental theorem in connexion with the representations, as an irreducible group of linear substitutions, of a group of finite order N is the following.

If r is the number of different sets of conjugate operations in the group, then, when the group of N permutations is completely reduced,

(i.) just r distinct irreducible representations occur:

(ii.) each of these occurs a number of times equal to the number of symbols on which it operates:

(iii.) these irreducible representations exhaust all the distinct irreducible representations of the group.

Among these representations what is called the "identical" representation necessarily occurs, i.e. that in which each operation of the group corresponds to leaving a single symbol unchanged. If these representations are denoted by [Gamma]1, [Gamma]2, ..., [Gamma]_r, then any representation of the group as a group of linear substitutions, or in particular as a group of permutations, may be uniquely represented by a symbol [Sigma][alpha]_i[Gamma]_i, in the sense that the representation when completely reduced will contain the representation [Gamma]_i just [alpha]_i times for each suffix i.

Group characteristics.

A representation of a group of finite order as an irreducible group of linear substitutions may be presented in an infinite number of equivalent forms. If

x'_i = [Sigma] s_(ij)x_j (i, j = 1, 2, ..., m),

is the linear substitution which, in a given irreducible representation of a group of finite order G, corresponds to the operation S, the determinant

| s11 - [lambda] s12 ... s_(1m) | | s21 s22-[lambda] ... s_(2m) | | . . ... . | | . . ... . | | . . ... . | | s_m1 s_2m ... s_(mm) - [lambda] |

is invariant for all equivalent representations, when written as a polynomial in [lambda]. Moreover, it has the same value for S and S', if these are two conjugate operations in G. Of the various invariants that thus arise the most important is s11 + s22 + ... + s_(mm), which is called the "characteristic" of S. If S is an operation of order p, its characteristic is the sum of m pth roots of unity; and in particular, if S is the identical operation its characteristic is m. If r is the number of sets of conjugate operations in G, there is, for each representation of G as an irreducible group, a set of r characteristics: X1, X2, ... X_r, one corresponding to each conjugate set; so that for the r irreducible representations just r such sets of characteristics arise. These are distinct, in the sense that if [Psi]1, [Psi]2, ..., [Psi]_r are the characteristics for a distinct representation from the above, then X_i and [Psi]_i are not equal for all values of the suffix i. It may be the case that the r characteristics for a given representation are all real. If this is so the representation is said to be self-inverse. In the contrary case there is always another representation, called the "inverse" representation, for which each characteristic is the conjugate imaginary of the corresponding one in the original representation. The characteristics are subject to certain remarkable relations. If h_p denotes the number of operations in the pth conjugate set, while X^_i{p}, and X^j{p} are the characteristics of the pth conjugate set in [Gamma]_i and [Gamma]_j, then

__p=r \ h_p X_p^i X^_p^j = 0 or n, /__p=1

according to [Gamma]_i and [Gamma]_j are not or are inverse representations, n being the order of G.

Again

__i=r \ X_p^i X^_q^i = 0 or n/h_p /__i=1

according as the pth and qth conjugate sets are not or are inverse; the qth set being called the inverse of the pth if it consists of the inverses of the operations constituting the pth.

Linear homogeneous groups.

Another form in which every group of finite order can be represented is that known as a linear homogeneous group. If in the equations

x'_r = a_(r1)x1 + a_(r2)x2 + ... +a_(rm)x_m, (r = 1, 2, ..., m),

which define a linear homogeneous substitution, the coefficients are integers, and if the equations are replaced by congruences to a finite modulus n, the system of congruences will give a definite operation, provided that the determinant of the coefficients is relatively prime to n. The product of two such operations is another operation of the same kind; and the total number of distinct operations is finite, since there is only a limited number of choices for the coefficients. The totality of these operations, therefore, constitutes a group of finite order; and such a group is known as a _linear homogeneous_ group. If n is a prime the order of the group is

(n^m - 1)(n^m - n) ... (n^m - n^(m-1)).

The totality of the operations of the linear homogeneous group for which the determinant of the coefficients is congruent to unity forms a subgroup. Other subgroups arise by considering those operations which leave a function of the variables unchanged (mod. n). All such subgroups are known as linear homogeneous groups.

When the ratios only of the variables are considered, there arises a _linear fractional_ group, with which the corresponding linear homogeneous group is isomorphic. Thus, if p is a prime the totality of the congruences

az + b z' [equiv] ------, ad - bc [/=] 0, (mod. p) cz + d

constitutes a group of order p(p^2 - 1). This class of groups for various values of p is almost the only one which has been as yet exhaustively analysed. For all values of p except 3 it contains a simple self-conjugate subgroup of index 2.

A great extension of the theory of linear homogeneous groups has been made in recent years by considering systems of congruences of the form

x'_r [equiv] a_(r1)x1 + a_(r2)x2 + ... + a_(rm)x_m, (r = 1, 2, ..., m),

in which the coefficients a_(rs), are integral functions with real integral coefficients of a root of an irreducible congruence to a prime modulus. Such a system of congruences is obviously limited in numbers and defines a group which contains as a subgroup the group defined by the same congruences with ordinary integral coefficients.

Applications.

The chief application of the theory of groups of finite order is to the theory of algebraic equations. The analogy of equations of the second, third and fourth degrees would give rise to the expectation that a root of an equation of any finite degree could be expressed in terms of the coefficients by a finite number of the operations of addition, subtraction, multiplication, division, and the extraction of roots; in other words, that the equation could be solved by radicals. This, however, as proved by Abel and Galois, is not the case: an equation of a higher degree than the fourth in general defines an algebraic irrationality which cannot be expressed by means of radicals, and the cases in which such an equation can be solved by radicals must be regarded as exceptional. The theory of groups gives the means of determining whether an equation comes under this exceptional case, and of solving the equation when it does. When it does not, the theory provides the means of reducing the problem presented by the equation to a normal form. From this point of view the theory of equations of the fifth degree has been exhaustively treated, and the problems presented by certain equations of the sixth and seventh degrees have actually been reduced to normal form.

Galois (see EQUATION) showed that, corresponding to every irreducible equation of the nth degree, there exists a transitive substitution-group of degree n, such that every function of the roots, the numerical value of which is unaltered by all the substitutions of the group can be expressed rationally in terms of the coefficients, while conversely every function of the roots which is expressible rationally in terms of the coefficients is unaltered by the substitutions of the group. This group is called the group of the equation. In general, if the equation is given arbitrarily, the group will be the symmetric group. The necessary and sufficient condition that the equation may be soluble by radicals is that its group should be a soluble group. When the coefficients in an equation are rational integers, the determination of its group may be made by a finite number of processes each of which involves only rational arithmetical operations. These processes consist in forming resolvents of the equation corresponding to each distinct type of subgroup of the symmetric group whose degree is that of the equation. Each of the resolvents so formed is then examined to find whether it has rational roots. The group corresponding to any resolvent which has a rational root contains the group of the equation; and the least of the groups so found is the group of the equation. Thus, for an equation of the fifth degree the various transitive subgroups of the symmetric group of degree five have to be considered. These are (i.) the alternating group; (ii.) a soluble group of order 20; (iii.) a group of order 10, self-conjugate in the preceding; (iv.) a cyclical group of order 5, self-conjugate in both the preceding. If x0, x1, x2, x3, x4 are the roots of the equation, the corresponding resolvents may be taken to be those which have for roots (i.) the square root of the discriminant; (ii.) the function (x0x1 + x1x2 + x2x3 + x3x4 + x4x0)(x0x2 + x2x4 + x4x1 + x1x3 + x3x0); (iii.) the function x0x1 + x1x2+ x2x3 + x3x4 + x4x0; and (iv.) the function x0^2x1 + x1^2x2 + x2^2x3 + x3^2x4 + x4^2x0. Since the groups for which (iii.) and (iv.) are invariant are contained in that for which (ii.) is invariant, and since these are the only soluble groups of the set, the equation will be soluble by radicals only when the function (ii.) can be expressed rationally in terms of the coefficients. If

(x0x1 + x1x2 + x2x3 + x3x4 + x4x0)(x0x2 + x2x4 + x4x1 + x1x3 + x3x0)

is known, then clearly x0x1 + x1x2 + x2x3 + x3x4 + x4x0 can be determined by the solution of a quadratic equation. Moreover, the sum and product (x0 + [epsilon]x1 + [epsilon]^2x2 + [epsilon]^3x3 + [epsilon]^4x4)^5 and (x0 + [epsilon]^4x1+[epsilon]^3x2 + [epsilon]^2x3 + [epsilon]x4)^5 can be expressed rationally in terms of x0x1 + x1x2 + x2x3 + x3x4 + x4x0, [epsilon], and the symmetric functions; [epsilon] being a fifth root of unity. Hence (x0 + [epsilon]x1 + [epsilon]^2x2 + [epsilon]^3x3 + [epsilon]^4X4)^5 can be determined by the solution of a quadratic equation. The roots of the original equation are then finally determined by the extraction of a fifth root. The problem of reducing an equation of the fifth degree, when not soluble by radicals, to a normal form, forms the subject of Klein's _Vorlesungen uber das Ikosaeder_. Another application of groups of finite order is to the theory of linear differential equations whose integrals are algebraic functions. It has been already seen, in the discussion of discontinuous groups in general, that the groups of such equations must be groups of finite order. To every group of finite order which can be represented as an irreducible group of linear substitutions on n variables will correspond a class of irreducible linear differential equations of the nth order whose integrals are algebraic. The complete determination of the class of linear differential equations of the second order with all their integrals algebraic, whose group has the greatest possible order, viz. 120, has been carried out by Klein.