chapter  6
44 Pages

Groups

WithGarrett Birkhoff, Saunders Mac Lane

The idea of "symmetry" is familiar to every educated person. But fewer people realize that there is a consequential algebra of symmetry. The symmetries of a cube form an interesting group. Geometrically speaking, these symmetries are the one-one transformations which preserve distances on the cube. They are known as "isometries," and are 48 in number. A familiar group containing an infinite number of transformations is the so-called Euclidean group. This consists of the "isometries" of the plane—or, in the language of elementary geometry, of the transformations under which the plane is congruent to itself. Felix Klein has eloquently described how the different branches of geometry can be regarded as the study of those properties of suitable spaces which are preserved under appropriate groups of transformations. Thus Euclidean geometry deals with those properties of space preserved under all isometries, and topology with those which are preserved under all homeomorphisms.