The Fitting length of finite soluble groups II

Jabara, Enrico

doi:10.1016/j.jalgebra.2017.06.002

Cited by 9 publications

(11 citation statements)

References 17 publications

(13 reference statements)

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“…However, there are some nice bounds proved in the case that the action is fixed point free, that is, when C G (A)=1: Under some further assumptions on the structure of A and G, a linear bound has been obtained for h(G) depending on l(A) in [2,3], and [8]. A current article established a quadratic bound h(G) ≤ 7l(A) 2 without any further assumptions on A or G (see [6,Corollary 1.2]). Since Theorem F creates a bridge between the general case and the fixed point free case, we might hope to attack the general case by using Theorem F and those results for the fixed point free case.…”

Section: Question 17 Under the Hypothesis Of Theorem E Is F(g A) Boun...mentioning

confidence: 98%

A Fitting height lemma and its applications

Kızmaz

2021

Arch. Math.

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Let A be a group acting on a solvable group G and let N be an We prove the inequality h([G, A]) ≤ h([G, A]N/N ) + h([N, A])where h(G) denotes the Fitting height of G. As an application of this result, we obtain several Fitting height inequalities. A new concept "fixed point free separability" and a new characteristic subgroup Y (G) is defined and used in order to prove some further results about the Fitting height of a group. In the last section, a new characterization of solvable groups is given: a group G is solvable if and only if it is fixed point free separable.

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Section: Question 17 Under the Hypothesis Of Theorem E Is F(g A) Boun...mentioning

confidence: 98%

A Fitting height lemma and its applications

Kızmaz

2021

Arch. Math.

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“…By Proposition 4.1, we have α(|ϕ | S |) 4 deg(f (x)). Then by Dade's theorem [4] the Fitting height of S is bounded in terms of α(|ϕ | S |), and Jabara's paper [17] gives the bound 7 • α(|ϕ | S |) 2 . So, for each such prime q, we obtain h(S) 112 • deg(f (x)) 2 and therefore h(G/O q ′ ,q (G)) 1 + 112 • deg(f (x)) 2 .…”

Section: Proof Note That Deg(f (X))mentioning

confidence: 99%

“…The first step in the proof of both Theorems 1.3 and 1.4 is to use Hall-Higman type theorems (with certain modifications) for obtaining a reduction to the situation where ϕ has order bounded in terms of deg(f (x)). Then the proof of Theorem 1.3 follows by an application of a special case of Dade's theorem [4], or rather Jabara's [17] recent improvement for the bound for the Fitting height of a finite group admitting a fixed-point-free automorphism of not necessarily coprime order.…”

Section: Introductionmentioning

confidence: 99%

Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity

Khukhro¹,

Moens²

2022

Preprint

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Let f (x) be a non-zero polynomial with integer coefficients. An automorphism ϕ of a group G is said to satisfy the elementary abelian identity f (x) if the linear transformation induced by ϕ on every characteristic elementary abelian section S of G is annihilated by f (x). We prove that if a finite (soluble) group G admits a fixed-point-free automorphism ϕ satisfying an elementary abelian identity f (x), where f (x) is a primitive polynomial, then the Fitting height of G is bounded in terms of deg(f (x)). We also prove that if f (x) is any non-zero polynomial and G is a σ ′ -group for a finite set of primes σ = σ(f (x)) depending only on f (x), then the Fitting height of G is bounded in terms of the number irr(f (x)) of irreducible factors in the decomposition of f (x). These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length α(|ϕ|) of ϕ when deg f (x) or irr(f (x)) is small in comparison with α(|ϕ|).

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“…Theorem 8.4.2 (Hoffman [25], Gross [19], Khukhro [27]; Jabara [31]). Consider a finite group G admitting a regular automorphism of order n, and let l n be the number of primes dividing n (counted with multiplicity).…”

Section: Split Polynomials (Fail)mentioning

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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The Fitting length of finite soluble groups II

Cited by 9 publications

References 17 publications

A Fitting height lemma and its applications

A Fitting height lemma and its applications

Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity

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

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