INVARIANTS OF A MAPPING OF A SET TO THE TWO-DIMENSIONAL EUCLIDEAN SPACE

Let E2 be the 2-dimensional Euclidean space and T be a set such that it has at least two elements. A mapping alpha : T -> E2 will be called a T-figure in E2. Let R be the field of real numbers and O(2, R) be the group of all orthogonal transformations of E2. Put SO(2, R) = {g is an element of O(2, R)|detg = 1}, MO(2, R) = {F : E2 -> E2| Fx= gx +b, g is an element of O(2, R), b is an element of E2}, MSO(2, R) = {F is an element of MO(2, R)|detg = 1}. The present paper is devoted to solutions of problems of G-equivalence of T-figures in E2 for groups G = O(2, R), SO(2, R), MO(2, R), MSO(2, R). Complete systems of G-invariants of T-figures in E2 for these groups are obtained. Complete systems of relations between elements of the obtained complete systems of G-invariants are given for these groups.