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zbMATH Open
Article . 1973
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Transactions of the American Mathematical Society
Article . 1973 . Peer-reviewed
Data sources: Crossref
Transactions of the American Mathematical Society
Article . 1973 . Peer-reviewed
Data sources: Crossref
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Classes of Automorphisms of Free Groups of Infinite Rank

Classes of automorphisms of free groups of infinite rank
Authors: Cohen, Robert;

Classes of Automorphisms of Free Groups of Infinite Rank

Abstract

This paper is concerned with finding classes of automorphisms of an infinitely generated free group F which can be generated by “elementary” Nielsen transformations. Two different notions of “elementary” Nielsen transformations are explored. One leads to a classification of the automorphisms generated by these transformations. The other notion leads to the subgroup B of Aut ( F ) {\operatorname {Aut}}(F) consisting of the “bounded length” automorphisms of F. We prove that the class of “bounded 3-length” automorphisms B 3 {B_3} and the class of “elementary simultaneous” Nielsen transformations generate the same subgroup of Aut ( F ) {\operatorname {Aut}}(F) . We show that for the class T of automorphisms of “2 occurring generators", the groups generated by T ∩ B T \cap B and the “elementary simultaneous” Nielsen transformations are identical. These results lead to the conjecture that B is generated by the “elementary simultaneous Nielsen transformations". A study is also made of the subgroup S of the “triangular automorphisms” of F ∞ {F_\infty } , the free group on a countably infinite set of free generators. It is found that a “triangular automorphism” may be factored into “splitting automorphisms” of F ∞ {F_\infty } , which may be viewed as the “elementary” automorphisms of S.

Keywords

Automorphisms of infinite groups, Free nonabelian groups

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
7
Average
Top 10%
Average
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