Geometry as an extension of the group theory

Abstract

Klein’s Erlangen program contains the postulate to study the group of automorphisms instead of a structure itself. This postulate, taken literally, sometimes means a substantial loss of information. For example, the group of automorphisms of the field of rational numbers is trivial. However in the case of Euclidean plane geometry the situation is different. We shall prove that the plane Euclidean geometry is mutually interpretable with the elementary theory of the group of authomorphisms of its standard model. Thus both theories differ practically in the language only.