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 \({\mathfrak{S}_6}\).