ferent from the group formed by the four roots of the equation . It is easy to prove that these two groups represent all the possible types of groups of four operations; that is, there are only two abstract groups of four operations. In general, the number of groups which can be formed with operations increases very rapidly with the number of factors of . When or 12 the number of possible abstract groups is 5.
Similarly, all the movements of space which transform a given solid into itself form a group. For instance, the cube is transformed into itself by twenty-four distinct movements. Nine around the lines which join the middle points of opposite faces, six around those which join the middle points of opposite edges, eight around the diagonals, and the identity. The group formed by these twenty-four movements is simply isomorphic with the one formed by the total number of permutations of four things. The regular octahedron has the same group, while the group of the regular tetrahedron is a subgroup of this group. The icosahedron and the duodecahedron have a common group of sixty operations. The groups of the regular solids play an important role in the theory of transformations of space. They are treated at considerable length in Klein's 'Ikosaeder' as well as in many other works.
All the preceding examples relate to groups of a finite number of operations, or of a finite order. During recent years the applications of groups of infinite order have been studied very extensively. As the theory of groups of finite order had its origin in the theory of algebraic equations, so the theory of groups of an infinite order might be said to have had its origin in the theory of differential equations. The rapid development of both of these theories is, however, due to the fact that much wider applications soon presented themselves. This is especially true of the latter. In fact, the earliest developments of the groups of infinite order were made without any view to their application to differential equations.
One of the simplest examples of groups of infinite order is furnished by the integral numbers when they are combined with respect to addition. The totality of the rational numbers clearly becomes a group when they are combined with respect to either of the operations addition or multiplication. The same remark applies evidently to all the real numbers as well as to all the complex numbers. These additive groups of infinite order are frequently represented by the equation , where may assume all the values of one of the given groups. If a may assume all real values the group is said to be continuous. When is restricted to rational values the group is said to be discontinuous, notwithstanding the fact that it transforms every finite point into a point which is indefinitely close to it.
