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 and the artinian-finitary automorphism group of M over R is . 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.