Automorphisms compatible with group actions

Abstract:

Let $A$ be a group of bijections of the set $\mathcal {V}$ and let $G$ be a subgroup of $A$. Our purpose here is to introduce several other subgroups of $A$ consisting of bijections compatible with the $G$-orbit structure on $\mathcal {V}$ and to investigate these groups when $\mathcal {V}$ is the Grassmann variety of all $p$-dimensional subspaces of a vector space $V$ and $G$ is a group of automorphisms of the Grassmannian induced by a group of linear transformations on $V$.