Characterizations of strong semilinear embeddings in terms of general linear and projective linear groups

Abstract

Let and be vector spaces over division rings. Suppose is finite and not less than . Consider a mapping with the following property: for every , there is such that . Our first result states that is a strong semilinear embedding if is non-constant and the dimension of the subspace of spanned by is not greater than . We present examples showing that these conditions cannot be omitted. Denote by the projective space associated with and consider the mapping with the following property: for every , there is such that . By the second result, is induced by a strong semilinear embedding of in if is non-constant and its image is contained in a subspace of whose dimension is not greater than , we also require that is a field.