__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.
AUTHORITIES.--Continuous groups: Lie and Engel, _Theorie der
Transformationsgruppen_ (Leipzig, vol. i., 1888; vol. ii., 1890; vol.
iii., 1893); Lie and Scheffers, _Vorlesungen uber gewohnliche
Differentialgleichungen mit bekannten infinitesimalen
Transformationen_ (Leipzig, 1891); _Idem, Vorlesungen uber
continuierliche Gruppen_ (Leipzig, 1893); _Idem, Geometrie der
Beruhrungstransformationen_ (Leipzig, 1896); Klein and Schilling,
_Hohere Geometrie_, vol. ii. (lithographed) (Gottingen, 1893, for both
continuous and discontinuous groups). Campbell, _Introductory Treatise
on Lie's Theory of Finite Continuous Transformation Groups_ (Oxford,
1903). Discontinuous groups: Klein and Fricke, _Vorlesungen uber die
Theorie der elliptischen Modulfunktionen_ (vol. i., Leipzig, 1890)
(for a full discussion of the modular group); _Idem, Vorlesungen uber
die Theorie der automorphen Funktionen_ (vol. i., Leipzig, 1897; vol.
ii. pt. i., 1901) (for the general theory of discontinuous groups);
Schoenflies, _Krystallsysteme und Krystallstruktur_ (Leipzig, 1891)
(for discontinuous groups of motions); Groups of finite order: Galois,
_[OE]uvres mathematiques_ (Paris, 1897, reprint); Jordan, _Traite des
substitutions et des equations algebriques_ (Paris, 1870); Netto,
_Substitutionentheorie und ihre Anwendung auf die Algebra_ (Leipzig,
1882; Eng. trans. by Cole, Ann Arbor, U.S.A., 1892); Klein,
_Vorlesungen uber das Ikosaeder_ (Leipzig, 1884; Eng. trans. by
Morrice, London, 1888); H. Vogt, _Lecons sur la resolution algebrique
des equations_ (Paris, 1895); Weber, _Lehrbuch der Algebra_
(Braunschweig, vol. i., 1895; vol. ii., 1896; a second edition
appeared in 1898); Burnside, _Theory of Groups of Finite Order_
(Cambridge, 1897); Bianchi, _Teoria dei gruppi di sostituzioni e delle
equazioni algebriche_ (Pisa, 1899); Dickson, _Linear Groups with an
Exposition of the Galois Field Theory_ (Leipzig, 1901); De Seguier,
_Elements de la theorie des groupes abstraits_ (Paris, 1904), A
summary with many references will be found in the _Encyklopadie der
mathematischen Wissenschaften_ (Leipzig, vol. i., 1898, 1899).
(W. Bu.)