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Let M be a module over the commutative ring R. The finitary automorphism group of M over R is
FAutRM = {g Î AutRM :M(g-1)isR-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
F1AutRM = {g Î AutRM : M(g-1)isR-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.

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Let M be any Abelian group. We make a detailed study for reasons explained in the Introduction of the normal subgroup
F¥ AutM = { g Î AutM:M(g - 1) is a minimax group}F_\infty AutM = \{ g \in AutM:M(g - 1) is a minimax group\}
of the automorphism group Aut M of M. The conclusions, although slightly weaker than one would hope, in that they do not fully explain the common behavior of
the finitary and the Artinian-finitary subgroups of Aut M, are certainly stronger than one might reasonably expect. Our main focus is on residual properties and unipotence.

1. Fundamental Concepts in the Theory of Infinite Groups.- 2. Soluble and Nilpotent Groups.- 3. Maximal and Minimal Conditions.- 4. Finiteness Conditions on Conjugates and Commutators.- 5. Finiteness Conditions on the Subnormal Structure of a Group.- Author Index.

In this chapter we are concerned with finiteness conditions which refer only to the normal or subnormal subgroups of a group. Now a simple group has only two subnormal subgroups and its subnormal structure is therefore very obvious; yet a simple group may have exceedingly complex general structure—as we will show in the first section of this chapter. For this reason we will often investigate finiteness conditions referring to the normal or subnormal structure of a group in conjunction with an additional property such as generalized solubility of a suitable type.

In this paper we attempt to describe the structure of groups G of automorphisms of an abelian group M with the property that M(g - 1) is finite for every element g of G. These groups are closely related to the finitary linear groups over finite fields. The abelian case is critical for our work and the core result of this paper is the following. An abelian group A is isomorphic to a group G as above with M torsion if and only if A is torsion and has a residually-finite subgroup B with A/B a direct sum of cyclic groups.

We introduce a wide range of generalized finitary automorphism groups of an arbitrary module M over an arbitrary ring R. The largest such subgroup of AutRM that we seriously consider here is the subgroup of all R-automorphisms g of M such that M(g − 1) has Krull dimension. We also consider the subgroup of all R-automorphisms g of M such that M(g − 1) is Artinian as an R-module. The results are vaguely analogous to the genuine finitary case but are somewhat weaker.

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.

In this paper we attempt to describe the structure of groups G of automorphisms of an abelian group M with the property that M(g - 1) is finite for every element g of G. These groups are closely related to the finitary linear groups over finite fields. The abelian case is critical for our work and the core result of this paper is the following. An abelian group A is isomorphic to a group G as above with M torsion if and only if A is torsion and has a residually-finite subgroup B with A/B a direct sum of cyclic groups.

Let M be a module over the ring R. The finitary automorphism group of M over R is and the Artinian-finitary automorphism group of M over R is We continue our study begun in [ 9 ] and [ 12 ] of the relationship between these two in some ways similar, but actually quite different types of automorphism group. Our study here involves the generalized finitary group $\{{\rm g} \in {\rm Aut}_{\rm R}{\rm M}{:}\,{\rm M}({\rm g} - 1)$ embeds into some finitely generated R-module which does not fit our earlier pattern of generalized finitary groups as described in [ 8 ].

In a series of papers, we have considered finitary (that is, Noetherian-finitary) and Artinian-finitary groups of automorphisms of arbitrary modules over arbitrary rings. The structural conclusions for these two classes of groups are really very similar, especially over commutative rings. The question arises of the extent to which each class is a subclass of the other.
Here we resolve this question by concentrating just on the ground ring of the integers ℤ. We show that even over ℤ neither of these two classes of groups is contained in the other. On the other hand, we show how each group in either class can be built out of groups in the other class. This latter fact helps to explain the structural similarity of the groups in the two classes.

We introduce a theory of finitary automorphism groups of arbitrary modules over arbitrary commutative rings, encompassing the theory of finitary linear groups (the field and vector space case). In particular, we have extended the structure theorems of U. Meierfrankenfeld et al. (J. London Math. Soc. 47 (1993 ) 31–40) for locally soluble finitary groups and unipotent finitary groups from the field to the commutative ring case.

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Artinian-finitary groups are locally normal-finitary. J Algebra (to appear) Author's address

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Wehrfritz BAF (2005) Artinian-finitary groups are locally normal-finitary. J Algebra (to appear) Author's address: B. A. F. Wehrfritz, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, England, e-mail: b.a.f.wehrfritz@qmul.ac.uk The Similarity Between Finitary and Artinian-Finitary Groups

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F 1994Hypercentral unipotent skew linear groupsComm Algebra22929949MATHMathSciNet

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