The Maximality of the Group of Euclidean Similarities within the Affine Group

Abstract

The purpose of this note is to show by elementary means that over the field of real numbers, or more generally over any Euclidean field K with Archimedean order the group of n-dimensional Euclidean similarities is maximal within the group of all affine mappings having a determinant of the form $ \pm λⁿ ≠ 0$. As a corollary it turns out that the orthogonal group $O_{n}(K)$ is maximal within the group ${SL_{n}(K)} ^\pm$ of all matrices of determinant $ \pm 1$.