Linear Groups with Bounded Action

Abstract

If R is a ring, G a group and A a right RG-module, then G has boundedly finite orbits on A if there is a positive integer b such that for all a ∈ A, the set {ag | g ∈ G} has order at most b. We study groups G with this property, proving that G/CG(A) is abelian-by-finite and that it is quite often the case that G/CG(A) is finite.