2010
DOI: 10.1007/s00013-009-0084-6
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On almost regular automorphisms

Abstract: Among other things, we prove that a polycyclic group admitting an automorphism of order two with finitely many fixed elements is abelian-by-finite. An example shows that this result cannot be extended to finitely generated soluble groups.

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Cited by 5 publications
(9 citation statements)
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“…For example, in the infinite dihedral group D = a, b | a 2 = b 2 = 1 , the automorphism ϕ defined by ϕ(a) = b and ϕ(b) = a is regular of order 2 but D is not nilpotent. In [2], we proved that a polycyclic group admitting an almost regular automorphism of order 2 is abelian-by-finite. Here we give a more general form of this result.…”
Section: Introduction and Main Resultsmentioning
confidence: 98%
See 3 more Smart Citations
“…For example, in the infinite dihedral group D = a, b | a 2 = b 2 = 1 , the automorphism ϕ defined by ϕ(a) = b and ϕ(b) = a is regular of order 2 but D is not nilpotent. In [2], we proved that a polycyclic group admitting an almost regular automorphism of order 2 is abelian-by-finite. Here we give a more general form of this result.…”
Section: Introduction and Main Resultsmentioning
confidence: 98%
“…Observe that Theorem 1.1 cannot be extended to the class of finitely generated soluble groups, even metabelian. Indeed the restricted wreath product Z Z has a regular automorphism of order 2 (see [2]), although this group is not nilpotent-by-finite.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
See 2 more Smart Citations
“…There is no advantage for the above theorems in assuming that the automorphism φ is actually fixed-point-free. As pointed out in [3,2] the infinite dihedral group is polycyclic but not nilpotent and the wreath product of two infinite cyclic groups is metabelian but not nilpotent-by-finite and yet both have fixed-point-free automorphisms of order 2. The latter group is isomorphic to a linear group of degree 2, but only over a field of positive transcendence degree (and characteristic zero).…”
Section: Corollary 1 Let G Be a Finite Extension Of A Torsion-free Smentioning
confidence: 98%