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- W2795492045 abstract "Let $V$ and $V'$ be vector spaces over division rings. Suppose $dim V$ is finite and not less than 3. Consider a mapping $l:Vto V$ with the following property: for every $uin {rm GL}(V)$ there is $u'in {rm GL}(V')$ such that $lu=u'l$. Our first result states that $l$ is a strong semilinear embedding if $l|_{Vsetminus{0}}$ 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 $hin {rm PGL}(V)$ 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." @default.
- W2795492045 created "2018-04-13" @default.
- W2795492045 creator A5060769535 @default.
- W2795492045 date "2012-07-16" @default.
- W2795492045 modified "2023-09-23" @default.
- W2795492045 title "Characterizations of strong semilinear embeddings" @default.
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