Algebraic Groups Whose Orbit Closures Contain Only Finitely Many Orbits

We explore connected affine algebraic groups G, which enjoy the following finiteness property (F): for every action of closure G-orbit contains only finitely many G-orbits. obtain two main results. First, we classify such groups. Namely, prove that a group G enjoys (F) if and is either torus or product one-dimensional unipotent group. Secondly, characterization in terms modality sense V. Arnol’d. irreducible variety X endowed with an equal to dim − maxxϵX G∙x.