Artinian-finitary groups are locally normal-finitary

Let M be a module over the commutative ring R. The finitary automorphism group of M over R is FAutRM={g∈AutRM:M(g−1) is R-noetherian} and the artinian-finitary automorphism group of M over R is F1AutRM={g∈AutRM:M(g−1) is R-artinian}. We investigate further the very close relationship between these two types of automorphism groups. The most interesting result in this present paper is the following. The group G=F1AutRM is locally normal-finitary; specifically every finite subset of G lies in a normal subgroup of G that is isomorphic to a finitary group of automorphisms of some module over some commutative ring.