Free-by-finite cyclic automorphism groups
type: Item
instance of: scholarly article
title: Free-by-finite cyclic automorphism groups
zbMATH Open document ID: 0782.20033
author: Martin R. Pettet
published in: Rocky Mountain Journal of Mathematics
publication date: 1994-03-10T00:00:00Z
review text: This paper is a contribution to the search of finding groups which can be or cannot be automorphism groups of other groups. Motivated by a result of J. L. Dyer that the group \(SL_ 2(Z)\) is not the automorphism group of any finitely generated group, he determines the structure of the group \(G\) whose automorphism group \(A\) is isomorphic to the fundamental group of a graph of locally cyclic groups. He, roughly, proves that \(G\) is infinitely generated abelian and \(A\) is free-by-finite cyclic with all torsion elements having order dividing 8, 10, 12 or 30. One must note the corollaries: (1) With the exception of the finite cyclic and infinite dihedral groups, the fundamental group of a graph of locally cyclic groups is not the automorphism group of any finitely generated group. (2) Except for the infinite dihedral group, a nontrivial free-by-finite cyclic group whose center contains no element of order 2 is not the automorphism group of any group. (3) An infinite free-by-finite cyclic group which contains an element of finite order not dividing 8, 10, 12 or 30 is not the automorphism group of any group. For the proof the author uses the Bass-Serre theory of groups acting on trees.
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 20E06
zbMATH DE Number: 218647
zbMATH Keywords: automorphism groups
zbMATH Keywords: fundamental group
zbMATH Keywords: graph of locally cyclic groups
zbMATH Keywords: torsion elements
zbMATH Keywords: free-by-finite cyclic group
zbMATH Keywords: groups acting on trees
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072625
OpenAlex ID: W2072872776
cites work: Every Countable Reduced Torsion-Free Ring is an Endomorphism Ring
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cites work: On solving the equation Aut(X)=G
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cites work: ALMOST-NILPOTENT PERIODIC GROUPS AS AUTOMORPHISM GROUPS
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DOI: 10.1216/RMJM/1181072625
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description: scientific article; zbMATH DE number 218647