On the centralizer and the commutator subgroup of an automorphism

Endimioni, Gérard; Moravec, Primož

doi:10.1007/s00605-011-0298-0

Cited by 7 publications

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“…Let α be an automorphism of a group G. Obviously, the centralizer C G (α) and the commutator subgroup [G, α] = g −1 g α ; g ∈ G are normal α-invariant subgroups of G. It is well known that the centralizer C G (α) in some sense has consequences on G, and in particular on G/[G, α]. For example, Endimioni and Moravec [3] proved that if α is an automorphism of a polycyclic group G and C G (α) is finite, then G/[G, α] is finite.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Polycyclic groups with automorphisms of order four

Zhou

Liu

2016

Czech Math J

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Polycyclic groups with automorphisms of order four

Zhou

Liu

2016

Czech Math J

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“…Endimioni and Moravec in [2] prove that if φ is an automorphism of a polycyclic group G with its fixed-point set…”

Section: Mathematics Subject Classification: 20f16 20e36mentioning

confidence: 99%

“…Endimioni and Moravec in [2] prove that if φ is an automorphism of a polycyclic group G with its fixed-point set C G (φ) finite, then G modulo [G, φ] = g −1 · gφ : g ∈ G is also finite (so [G, φ] is large), but the converse is false, even if φ has order 2, C G (φ) = 1 and G is polycyclic and metabelian. These and related results are extended in [9] to, in particular, soluble groups of finite rank.…”

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“…

Abstract: Let φ be an automorphism of a group G. Under various finiteness or solubility hypotheses, for example under polycyclicity, the commutator subgroup [G, φ] has finite index in G if the fixed-point set C G (φ) of φ in G is finite, but not conversely, even for polycyclic groups G. Here we consider a stronger, yet natural, notion of what it means for [G, φ] to have 'finite index' in G and show that in many situations, including G polycyclic, it is equivalent to C G (φ) being finite.

Mathematics Subject Classification: 20F16, 20E36.Key words: automorphism, fixed-point set, soluble group.Endimioni and Moravec in [2] prove that if φ is an automorphism of a polycyclic group G with its fixed-point setis large), but the converse is false, even if φ has order 2, C G (φ) = 1 and G is polycyclic and metabelian. These and related results are extended in [9] to, in particular, soluble groups of finite rank.

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On the fixed-point set of an automorphism of a group

Wehrfritz¹

2013

Publicacions Matemàtiques

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Let φ be an automorphism of a group G. Under various finiteness or solubility hypotheses, for example under polycyclicity, the commutator subgroup [G, φ] has finite index in G if the fixed-point set C G (φ) of φ in G is finite, but not conversely, even for polycyclic groups G. Here we consider a stronger, yet natural, notion of what it means for [G, φ] to have 'finite index' in G and show that in many situations, including G polycyclic, it is equivalent to C G (φ) being finite. Mathematics Subject Classification: 20F16, 20E36.Key words: automorphism, fixed-point set, soluble group.Endimioni and Moravec in [2] prove that if φ is an automorphism of a polycyclic group G with its fixed-point setis large), but the converse is false, even if φ has order 2, C G (φ) = 1 and G is polycyclic and metabelian. These and related results are extended in [9] to, in particular, soluble groups of finite rank. Is there some stronger notion of [G, φ] being large such that in these situations C G (φ) is finite if and only is [G, φ] is large in this sense? We will see below that the answer to this is yes.Let φ be an automorphism of a group G and define the map γ if G into itself by gγ = [g, φ] = g −1 · gφ. Then ker γ = {g ∈ G : gγ = 1} is C G (φ) and assumptions on C G (φ) should give information about Gγ. The problem is that γ is not usually a homomorphism and Gγ is not usually a subgroup of G. If S is any subset of G say that S has finite index in G if S contains a subgroup of G (normal if you wish) of finite index in G. If Gγ has finite index in G then so does [G, φ], since [G, φ] = Gγ . We shall see that in suitable situations C G (φ) is finite if and only if Gγ has finite index in G, and these situations include polycyclic groups and soluble groups of finite rank.We start by defining the classes of group we shall be mainly considering. A group G has finite Hirsch number if it has a series of finite

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“…Trivially G has Hirsch number 0. As Endimioni and Moravec point out in [1], we do need f of finite order in iii), even if G is free abelian of rank 2. Note that iii) implies that G=G; f is isomorphic to a finite extension of a section of C G (f); see Remark 6 below.…”

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On the Fixed-Point Set and Commutator Subgroup of an Authomorphism of a Group of Finite Rank

Wehrfritz¹

2012

Rend. Sem. Mat. Univ. Padova

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Let f be an automorphism of a group G. For G polycyclic, Endimioni and Moravec in [1] discuss the relationship between the fixed-point set C G (f) and the commutator subgroup [G, f] of f in G. Here we extend these results to soluble groups satisfying various rank restrictions. Suppose f is an automorphism of the polycyclic group G. Endimioni and Moravec in [1] prove the following. i) If C G (f) is finite and f has order 2, then [G, f] H is finite. ii) If C G (f) is finite, then so is G=G; f]. iii) If jfj is finite, then so is the index (G : [G, f].C G (f)). Of course C G (f) denotes the set of fixed points of f in G, [G, f] = h g À1 :gf : g P Gi and [G, f] is a f-invariant normal subgroup of G. Here we generalize these results to groups with some sort of rank restriction. Generally our groups at least have finite Hirsch number and satisfy something weaker than the FAR condition. We must start by defining these terms. If a group G has a series of finite length each factor of which is either locally finite or infinite cyclic, then the number of infinite cyclic factors in such a series is an invariant of G, which we call the Hirsch number of G (it is also sometimes called the torsion-free rank of G). This is the weakest of the rank restrictions considered here. The structure of groups with finite Hirsch number is discussed in many places (e.g. see [3]). From our point of view the most convenient description of the structure of such a group is

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On the centralizer and the commutator subgroup of an automorphism

Cited by 7 publications

References 6 publications

Polycyclic groups with automorphisms of order four

Polycyclic groups with automorphisms of order four

On the fixed-point set of an automorphism of a group

On the Fixed-Point Set and Commutator Subgroup of an Authomorphism of a Group of Finite Rank

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