A criterion for metanilpotency of a finite group

Bastos, Raimundo; Monetta, Carmine; Shumyatsky, Pavel

doi:10.1515/jgth-2018-0002

Cited by 11 publications

(13 citation statements)

References 4 publications

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“…Two easier counterexamples are given in Section 4. On the other hand, the answer to the above question is positive when w is a lower central word [1,2]. Motivated by this, we prove the following nilpotency criterion for w(G), where w is a simple commutator word without any repetition of the first variable.…”

Section: Introductionmentioning

confidence: 94%

A Nilpotency Criterion for Some Verbal subgroups

Monetta¹,

Tortora

2019

Bull. Aust. Math. Soc.

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The word w = [x i1 , x i2 , . . . , x i k ] is a simple commutator word if k ≥ 2, i 1 = i 2 and i j ∈ {1, . . . , m}, for some m > 1. For a finite group G, we prove that if i 1 = i j for every j = 1, then the verbal subgroup corresponding to w is nilpotent if and only if |ab| = |a||b| for any w-values a, b ∈ G of coprime orders. We also extend the result to a residually finite group G, provided that the set of all w-values in G is finite.

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Section: Introductionmentioning

confidence: 94%

A Nilpotency Criterion for Some Verbal subgroups

Monetta¹,

Tortora

2019

Bull. Aust. Math. Soc.

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“…Recall that a group G is metanilpotent if it has a normal subgroup N such that both N and G/N are nilpotent. The following lemma is well-known (see for example [3,Lemma 3]). Here F (K) denotes the Fitting subgroup of a group K and O p (K) stands for the maximal normal p -subgroup of K. We are now in a position to complete the proof of Theorem 1.1.…”

Section: Lemma 24 Let G Be a Finite Group And Let P Be A Sylow P-smentioning

confidence: 99%

“…Of course, the conditions in both above results are also necessary for the nilpotency of G and G , respectively. More recently, in [3], the above results were extended as follows.…”

mentioning

confidence: 91%

“…Of course, this is the familiar kth term of the lower central series of G. If k = 2, we have γ k (G) = G . The main result in [3] says that the kth term of the lower central series of a finite group G is nilpotent if and only if |ab| = |a||b| for any γ k -values a, b ∈ G of coprime orders. It was further conjectured in [3] that a similar criterion of nilpotency of the kth term of the derived series of G holds.…”

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confidence: 99%

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On nilpotency of higher commutator subgroups of a finite soluble group

Alves

Shumyatsky

2020

Arch. Math.

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Let G be a finite soluble group and G (k) the kth term of the derived series of G. We prove that G (k) is nilpotent if and only if |ab| = |a||b| for any δ k-values a, b ∈ G of coprime orders. In the course of the proof, we establish the following result of independent interest: let P be a Sylow p-subgroup of G. Then P ∩G (k) is generated by δ k-values contained in P (Lemma 2.5). This is related to the so-called focal subgroup theorem.

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“…The work of Baumslag and Wiegold has been extended to the realm of group-words in a series of papers [3,4,5,8,12]. In a similar matter, the aim of this work is to generalize Theorem A obtaining a verbal version.…”

Section: Introductionmentioning

confidence: 99%

$p$-nilpotency criteria for some verbal subgroups

Rojas¹,

Grazian²,

Monetta³

2021

Preprint

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Let G be a finite group, let p be a prime and let w be a group-word. We say that G satisfies P (w, p) if the prime p divides the order of xy for every w-value x in G of p ′ -order and for every non-trivial w-value y in G of order divisible by p. With k ≥ 2, we prove that the kth term of the lower central series of G is p-nilpotent if and only if G satisfies P (γ k , p). In addition, if G is soluble, we show that the kth term of the derived series of G is p-nilpotent if and only if G satisfies P (δ k , p).Here o(x) denotes the order of the group element x. We point out that, instead of studying the couple of elements (x, y), where x is a p ′ -element of prime power order and y is a non-trivial p-element, one can focus on the couple (x, y), where x is a p ′ -element and y is a non-trivial element of order divisible by p. Corollary B.Let G be a finite group and let p be a prime. Then G is p-nilpotent if and only if for every x ∈ G such that p does not divide o(x) and for every 1 = y ∈ G such that p divides o(y), p divides o(xy).

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A criterion for metanilpotency of a finite group

Abstract: We prove that the kth term of the lower central series of a finite group G is nilpotent if and only if |ab| = |a||b| for any γ k -commutators a, b ∈ G of coprime orders.2010 Mathematics Subject Classification. 20D30, 20D25.

Cited by 11 publications

References 4 publications

A Nilpotency Criterion for Some Verbal subgroups

A Nilpotency Criterion for Some Verbal subgroups

On nilpotency of higher commutator subgroups of a finite soluble group

$p$-nilpotency criteria for some verbal subgroups

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