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

Let $V$ and $V^{\mathrm{\prime }}$ 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 $u\in \mathrm{G}\mathrm{L}(V)$ there is $u^{\mathrm{\prime }}\in \mathrm{G}\mathrm{L}(V^{\mathrm{\prime }})$ such that $lu=u^{\mathrm{\prime }}l$. Our first result states that $l$ is a strong semilinear embedding if $l{\mathrm{\mid }}_{V\setminus 0}$ is non-constant and the dimension of the subspace of $V^{\mathrm{\prime }}$ 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 ${\mathcal P}(V)$theprojectivespaceassociatedwith$V$andconsiderthemapping$f:{\mathcal P}(V)\to {\mathcal P}(V’)$withthefollowingproperty:forevery$h\in {\rm PGL}(V)$thereis$h’\in {\rm PGL}(V’)$suchthat$fh=h’f$. By the second result, $f$ is induced by a strong semilinear embedding of $V$ in $V^{\mathrm{\prime }}$ if $f$ is non-constant and its image is contained in a subspace of $V^{\mathrm{\prime }}$ whose dimension is not greater than $n$, we also require that $R^{\mathrm{\prime }}$ is a field.