The Similarity Between Finitary and Artinian-Finitary Groups

Abstract

Let M be a module over the commutative ring R. The finitary automorphism group of M over R is \({\rm FAut}_RM =\{g\in{\rm Aut}_RM :M(g-1)\ {\rm is}\ R\hbox{-}{\rm Noetherian}\}\) and the Artinian-finitary automorphism group of M over R is \({\rm F}_1{\rm Aut}_RM = \{g\in{\rm Aut}_RM : M(g-1)\ {\rm is}\ R\hbox{-}{\rm Artinian}\}.\) We investigate further the surprisingly close relationship between these two types of automorphism groups. Their group theoretic properties seem practically identical.