We study the group IAut(A) generated by the inertial automorphisms of an abelian group A, that is, automorphisms f with the property that each subgroup H of A has finite index in the subgroup generated by H and f(H) . Clearly, IAut(A) contains the group FAut(A) of finitary automorphisms of A, which is known to be locally finite. In a previous paper, we showed that IAut(A) is (locally finite)-by-abelian. In this paper, we show that IAut(A) is also metabelian-by-(locally finite). More precisely, IAut(A) has a normal subgroup N such that IAut(A) /N is locally finite and the derived subgroup of N 0 is an abelian periodic subgroup all of whose subgroups are normal in N. In the case when A is periodic, IAut(A) turns out to be abelian-by-(locally finite) indeed, while in the general case it is not even (locally nilpotent)-by-(locally finite). Moreover, we provide further details about the structure of IAut(A) .
A group of generalized finitary automorphisms of an abelian group / Dardano, Ulderico; Rinauro, Silvana. - In: JOURNAL OF GROUP THEORY. - ISSN 1433-5883. - 20 (2017):(2017), pp. 347-369. [10.1515/jgth-2016-0043]
A group of generalized finitary automorphisms of an abelian group
DARDANO, ULDERICO;
2017
Abstract
We study the group IAut(A) generated by the inertial automorphisms of an abelian group A, that is, automorphisms f with the property that each subgroup H of A has finite index in the subgroup generated by H and f(H) . Clearly, IAut(A) contains the group FAut(A) of finitary automorphisms of A, which is known to be locally finite. In a previous paper, we showed that IAut(A) is (locally finite)-by-abelian. In this paper, we show that IAut(A) is also metabelian-by-(locally finite). More precisely, IAut(A) has a normal subgroup N such that IAut(A) /N is locally finite and the derived subgroup of N 0 is an abelian periodic subgroup all of whose subgroups are normal in N. In the case when A is periodic, IAut(A) turns out to be abelian-by-(locally finite) indeed, while in the general case it is not even (locally nilpotent)-by-(locally finite). Moreover, we provide further details about the structure of IAut(A) .File | Dimensione | Formato | |
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