On \(p\)-semilinear transformations (Q459254)

Let \(\mathbb F\) be a field of positive characteristic \(p\). A mapping \(f : V \rightarrow W\), where \(V\) and \(W\) are finite-dimensional vector spaces over \(\mathbb F\), is called \(p\)-semilinear if \(fk\alpha +\beta = kpf(\alpha)+f\beta\) for all \(\alpha, \beta \in V\) and \(k\in \mathbb{F}\). These occur in the theory of Mathieu groups and projective geometry, but their properties have mostly been studied for the case of characteristic \(0\). The authors give the following example for positive characteristic \(p\): the mapping \(f : \mathbb F \rightarrow \mathbb F\) is given by \(fx= xp\). They then prove some elementary properties of \(p\)-semilinear mappings, define their matrix with respect to a given basis and prove basis change and other results, many of which differ from the linear case. In particular, they prove a rank-nullity theorem and a result closely related to the Jordan-Chevalley decomposition.