There exist only a finite number of non-iso morphic types of finite groups of linear trans formations of z. If s is represented on a spher ical surface, every rotation of the sphere pro duces a linear transformation of z. Those ro tations of the sphere which convert into itself a regular solid inscribed in the sphere, or a regular polygon of n sides inscribed in a great circle (equator), form a group. These groups are of orders 60 (icosahedron, dodecahedron), 24 (octahedron, cube), 12 (tetrahedron), 2n (dihedron), n (cyclical). They give all the non-isomorphic types of finite groups of linear transformations of z. The octohedral group is also isomorphic with the symmetric substitution group of four letters, the tetrahedral and icosa hedral groups with (alternating) substitution groups of four and five letters, respectively.
A simple example of a (dihedral) group of order 6 is generated by the transformations S and T above.
The linear transformations of z written in homogeneous form s'r = as, fizr, = ys, dz, furnish homogeneous linear groups. Increas ing the number of variables, we arrive at the general homogeneous linear groups Si = au& + . . au au, Z, =an ar + ars as + . . . ... Jo' = Omit aneS, + • • • • + amen identified, for example, with project ive geometry. Curves, surfaces, etc., frequently have linear transformations into themselves, these always forming a group. Thus a plane cubic curve has in general such a group of order 216. Linear congruence groups should also be mentioned. An example is the simple group of order (p >3) composed of the lin ear transformation (a z / (y s + when a, 11, y, d, a', a are integers taken mod. p.
Continuous These are groups of transformations involving continuous para meters, such as the entire group of linear trans formations of z, or the entire group of motions in a plane or in space. The theory of these
groups, which has been extensively developed by Sophus Lie and his followers since 1870, has important applications to geometry, and es pecially to the theory of differential equations.
Historical.—The theory of groups was orig inally developed by Galois, Cauchy and their successors under the particular guise of substi tution groups. It was with Sylow's memoir in the 'Mathematische Annalen,> Vol. V (1872) that the theory began to assume its independent abstract form. Among those who contributed to this movement are Cayley, Klein, Dyck and others. But it is to Frobenius, above all others, that we owe the great developments of the pure theory which have been accomplished in the last 25 years. The theory of group characteristics, recently created by Frobenius, is destined to produce brilliant results in the near future.
Other historical elements are traceable in the accompanying bibliography.
Bibliography.— Burnside, 'Theory of Groups' (1897, second edition 1911) ; Cayley, 'American Journal of Mathematics' (1878) Dickson, 'Linear Groups' (1901) ; 'Theory of Equations' (1903) ; Easton, 'Constructive De velopment of Group Theory, with a Bibliog raphy' (1902) ; Frobenius, 'Sitzungsberichte' of the Berlin Academy (1895 et seq.) ; Jordan, 'Trait" des substitutions' (1871) ; Klein, 'Iko seeder' (1884) ; Retto, 'Substitutionentheorie' (1882) ; Serret, 'Algebre (1866); Sylow, 'Mathematische Annalen' (1872); Weber, 'Algebra' (1899).