Abstract
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.
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REFERENCES
B. A. F. Wehrfritz, “Finitary automorphism groups over commutative rings,” J. Pure Appl. Algebra (to appear).
B. A. F. Wehrfritz, “Nilpotence in finitary groups over commutative rings,” Commun. Algebra (to appear).
B. A. F. Wehrfritz, “Finitary automorphism groups over noncommutative rings,” J. London Math. Soc. (to appear).
B. A. F. Wehrfritz, “On generalized finitary groups,” J. Algebra (to appear).
B. A. F. Wehrfritz, Artinian-Finitary Groups over Commutative Rings, Preprint (2001).
D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Vol. 1, Springer, Berlin (1972).
D. W. Sharpe and P. Vámos, Injective Modules, Cambridge University Press, Cambridge (1972).
O. H. Kegel and B. A. F. Wehrfritz, Locally Finite Groups, North-Holland, Amsterdam (1973).
F. Leinen, “Hypercentral unipotent finitary skew linear groups,” Commun. Algebra, 22, 929–949 (1994).
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Wehrfritz, B.A.F. Finitary and Artinian-Finitary Groups over the Integers ℤ. Ukrainian Mathematical Journal 54, 924–936 (2002). https://doi.org/10.1023/A:1021756002353
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DOI: https://doi.org/10.1023/A:1021756002353