The affine preservers of non-singular matrices

Abstract

When ${\mathbb{K}}$ is an arbitrary field, we study the affine automorphisms of ${{\rm M}_n(\mathbb{K})}$ that stabilize ${{\rm GL}_n(\mathbb{K})}$. Using a theorem of Dieudonné on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n > 2 or # ${\mathbb{K} > 2}$. We include a short new proof of the more general Flanders theorem for affine subspaces of ${{\rm M}_{p,q}(\mathbb{K})}$ with bounded rank. We also find that the group of affine transformations of ${{\rm M}_2(\mathbb{F}_2)}$ that stabilize ${{\rm GL}_2(\mathbb{F}_2)}$ does not consist solely of linear maps. Using the theory of quadratic forms over ${\mathbb{F}_2}$, we construct explicit isomorphisms between it, the symplectic group ${{\rm Sp}_4(\mathbb{F}_2)}$ and the symmetric group ${\mathfrak{S}_6}$.