WHAT IS GROUP THEORY? 373 



ferent from the group formed by the four roots of the equation x^ = 1. 

 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 n operations increases very rapidly with the number of 

 factors of n. When n ^ 8 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 impor- 

 tant 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 applica- 

 tion to differential equations. 



One of the simplest examples of groups of infinite order is fur- 

 nished 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 x ^^= x' -]- a, where a 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 a 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. 



