Invariants of a mapping of a set to the two-dimensional Euclidean space

Abstract

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