Polycyclic group admitting an almost regular automorphism of prime order

Endimioni, Gérard

doi:10.1016/j.jalgebra.2010.03.015

Cited by 9 publications

(6 citation statements)

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Section: Introductionsupporting

confidence: 72%

“…For example, if m is prime then G is a finite extension of a nilpotent group and if m = 4 then G is a finite extension of a centre-by-metabelian group. This extends the special cases where G is polycyclic, proved recently by Endimioni (2010); see [3].An automorphism φ of a group G is said to be almost fixed-point-free (a phrase we shorten to afpf) if its fixed-point set C G (φ) is finite. In [3] Endimioni proves that a polycyclic-by-finite group with an afpf automorphism of prime order is nilpotent-by-finite.…”

mentioning

confidence: 57%

“…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: 99%

“…This is because in the notation of Lemma 1 the kernel of γ | A , namely C A (φ) is finite, so (A : Aγ ) is finite, again by counting the infinite factors in series for A and Aγ ; hence K /A is finite by Lemma 1. In particular (Endimioni [3]…”

Section: Lemma 2 Assume the Notation Of Lemma 1 Suppose That A Is Tmentioning

confidence: 99%

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Almost fixed-point-free automorphisms of soluble groups

Wehrfritz

2011

Journal of Pure and Applied Algebra

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a b s t r a c tSuppose G is either a soluble (torsion-free)-by-finite group of finite rank or a soluble linear group over a finite extension field of the rational numbers. We consider the implications for G if G has an automorphism of finite order m with only finitely many fixed points. For example, if m is prime then G is a finite extension of a nilpotent group and if m = 4 then G is a finite extension of a centre-by-metabelian group. This extends the special cases where G is polycyclic, proved recently by Endimioni (2010); see [3].An automorphism φ of a group G is said to be almost fixed-point-free (a phrase we shorten to afpf) if its fixed-point set C G (φ) is finite. In [3] Endimioni proves that a polycyclic-by-finite group with an afpf automorphism of prime order is nilpotent-by-finite. Here we present a somewhat shorter alternative proof of this that also delivers the same conclusion for finite extensions of torsion-free soluble minimax groups and also for soluble-by-finite linear groups over finite extension fields of the rational numbers Q. Trivially polycyclic groups are (torsion-free)-by-finite, soluble, minimax and of finite rank and, less trivially (e.g. see [7], 1.2), (torsion-free)-by-finite soluble groups of finite rank have faithful finite-dimensional representations over the rationals Q. Neither reverse implication holds.If G is a finite soluble group with a fixed-point-free automorphism of prime order p, then G is nilpotent of class bounded by a function of p only-see Higman [4]; the least such function we denote here by h(p). (By a famous theorem of J.G. Thompson the solubility restriction is unnecessary, but we only require the soluble case.) Endimioni actually proves that if G is polycyclic-by-finite with an afpf automorphism of prime order p, then G has a nilpotent normal subgroup of finite index and class at most h(p). This also applies to our minimax, finite rank and linear cases. Below, Theorem 1 is our basic result and its proof delivers our proof of Endimioni's theorem. Theorem 2 we derive from Theorem 1 and Corollary 1 is immediate from Theorem 2 and [7], 1.2.

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Section: Introductionsupporting

confidence: 72%

mentioning

confidence: 57%

Section: Corollary 1 Let G Be a Finite Extension Of A Torsion-free Smentioning

confidence: 99%

Section: Lemma 2 Assume the Notation Of Lemma 1 Suppose That A Is Tmentioning

confidence: 99%

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Almost fixed-point-free automorphisms of soluble groups

Wehrfritz

2011

Journal of Pure and Applied Algebra

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“…Such automorphisms naturally appear in the study of almost-regular automorphisms of prime order by Bettio, Endimioni, Jabara, Wehrfritz, Zappa, and others [7,16]. The solvability of G was proven by Hughes-Thompson using the fundamental results of Hall and Higman about minimal polynomials of operators on finite-dimensional vector spaces [22].…”

Section: Introduction 1motivationmentioning

confidence: 99%

The nilpotency of finite groups with a fix-point-free automorphism satisfying an identity

Moens¹

2018

Preprint

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We generalise the positive solution of the Frobenius conjecture (by J. Thompson) and refinements thereof (by Higman, Kreknin, and Kostrikin). This allows us to also extend the positive solution of the restricted Burnside problem for prime exponents (by Kostrikin) and a generalisation of it (by E. Khukhro).We do this by studying the structure of groups that admit an automorphism with a prescribed polynomial identity. In fact, to each polynomial r(twe assign integer-valued invariants ι 1 and ι 2 , and we will prove that they satisfy the following property. Let G be a finite group with an automorphism α :By specialising r(t) to linear, cyclotomic or Anosov polynomials, we can also recover and extend a number of results in the literature. * MSC2010: 20D45 (automorphisms of groups), 20D15 (nilpotent groups), 17B70 (graded Lie algebras). † This work was supported by the Austrian Science Fund (FWF) grants: J − 3371 − N 25 ("Representations and gradings of solvable Lie algebras") and P 30842 − N 35 ("Infinitesimal Lie rings: gradings and obstructions").

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Groups Admitting an Automorphism of Prime Order with Finite Centralizer

Bettio¹,

Jabara

Wehrfritz

2013

Mediterr. J. Math.

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We consider a group G with an automorphism of ﬁnite, usu- ally prime, order. If G has ﬁnite Hirsch number, and also if G satisﬁes various stronger rank restrictions, we study the consequences and equivalent hypotheses of having only ﬁnitely many ﬁxed-points. In particular we prove that if a group G with ﬁnite Hirsch number h admits an automorphism ϕ of prime order p such that |CG (ϕ)| = n < ∞, then G has a subgroup of ﬁnite index bounded in terms of p, n and h that is nilpotent of p-bounded class

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Polycyclic group admitting an almost regular automorphism of prime order

Cited by 9 publications

References 14 publications

Almost fixed-point-free automorphisms of soluble groups

Almost fixed-point-free automorphisms of soluble groups

The nilpotency of finite groups with a fix-point-free automorphism satisfying an identity

Groups Admitting an Automorphism of Prime Order with Finite Centralizer

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