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 constitutes a group of order p(p2 − 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 ≡ ar1x1 + ar2x2 + ... + armxm, (r ＝ 1, 2, ..., m),

in which the coefficients ars, 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.

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 ) 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 x02x1 + x12x2 + x22x3 + x32x4 + x42x0. 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 + x1 + 2x2 + 3x3 + 4x4)5 and (x0 + 4x1 + 3x2 + 2x3 + x4)5 can be expressed rationally in terms of x0x1 + x1x2 + x2x3 + x3x4 + x4x0,, and the symmetric functions; being a fifth root of unity. Hence (x0 + x1 + 2x2 + 3x3 + 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 über 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.

—Continuous groups: Lie and Engel, Theorie der Transformationsgruppen (Leipzig, vol. i., 1888; vol. ii., 1890; vol. iii., 1893); Lie and Scheffers, Vorlesungen über gewöhnliche Differentialgleichungen mit bekannten infinitesimalen Transformationen (Leipzig, 1891); Idem, Vorlesungen über continuierliche Gruppen (Leipzig, 1893); Idem, Geometrie der Berührungstransformationen (Leipzig, 1896); Klein and Schilling, Höhere Geometrie, vol. ii. (lithographed) (Göttingen, 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 über die Theorie der elliptischen Modulfunktionen (vol. i., Leipzig, 1890) (for a full discussion of the modular group); Idem, Vorlesungen über 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, Œuvres mathématiques (Paris, 1897, reprint); Jordan, Traité des substitutions et des équations algébriques (Paris, 1870); Netto, Substitutionentheorie und ihre Anwendung auf die Algebra (Leipzig, 1882; Eng. trans. by Cole, Ann Arbor, U.S.A., 1892); Klein, Vorlesungen über das Ikosaeder (Leipzig, 1884; Eng. trans. by Morrice, London, 1888); H. Vogt, Leçons sur la résolution algébrique des équations (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 Séguier, Éléments de la théorie des groupes abstraits (Paris, 1904), A summary with many references will be found in the Encyklopädie der mathematischen Wissenschaften (Leipzig, vol. i., 1898, 1899).

GROUSE, a word of uncertain origin, now used generally by ornithologists to include all the “rough-footed” Gallinaceous birds, but in common speech applied almost exclusively, when used alone, to the Tetrao scoticus of Linnaeus, the Lagopus scoticus of modern systematists—more particularly called in English the red grouse, but till the end of the 18th century almost invariably spoken of as the Moor-fowl or Moor-game. The effect which this species is supposed to have had on the British legislature, and therefore on history, is well known, for it was the common belief that parliament always rose when the season for grouse-shooting began (August 12th); while according to the Orkneyinga Saga (ed. Jonaeus, p. 356; ed. Anderson, p. 168) events of some importance in the annals of North Britain followed from its pursuit in Caithness in the year 1157.

The red grouse is found on moors from Monmouthshire and Derbyshire northward to the Orkneys, as well as in most of the Hebrides. It inhabits similar situations throughout Wales and Ireland, but it does not naturally occur beyond the limits of the British Islands, and is the only species among birds peculiar to them. The word “species” may in this case be used advisedly (since the red grouse invariably “breeds true,” it admits of an easy diagnosis, and it has a definite geographical range); but scarcely any zoologist can doubt of its common origin with the willow-grouse, Lagopus albus (L. subalpinus or L. saliceti of some authors), that inhabits a subarctic zone from Norway across the 