The Similarity Between Finitary and Artinian-Finitary Groups

Let M be a module over the commutative ring R. The finitary automorphism group of M over R is FAut_RM = {g ? Aut_RM :M(g-1) is R-Noetherian}{\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 F₁Aut_RM = {g ? Aut_RM : M(g-1) is R-Artinian}.{\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.