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

Abstract

Let $V$ and $V'$ be vector spaces over division rings. Suppose $\dim V$ is finite and not less than 3. Consider a mapping $l:V\to V$ with the following property: for every $u\in {\rm GL}(V)$ there is $u'\in {\rm GL}(V')$ such that $lu=u'l$. Our first result states that $l$ is a strong semilinear embedding if $l|_{V\setminus{0}}$ is non-constant and the dimension of the subspace of $V'$ spanned by $l(V)$ is not greater than $n$. We present examples showing that these conditions can not be omitted. In some special cases, this statement can be obtained from Dicks and Hartley (1991) and Zha (1996). Denote by ${\mathcal P}(V)$ the projective space associated with $V$ and consider the mapping $f:{\mathcal P}(V)\to {\mathcal P}(V')$ with the following property: for every $h\in {\rm PGL}(V)$ there is $h'\in {\rm PGL}(V')$ such that $fh=h'f$. By the second result, $f$ is induced by a strong semilinear embedding of $V$ in $V'$ if $f$ is non-constant and its image is contained in a subspace of $V'$ whose dimension is not greater than $n$, we also require that $R'$ is a field.