On commutators in groups

Kappe, Luise-Charlotte; Morse, Robert Fitzgerald

doi:10.1017/cbo9780511721205.018

Cited by 23 publications

(21 citation statements)

References 47 publications

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“…We will refer to Thompson's conjecture later in the text. For a detailed information about the status of these conjectures see the survey article [12]. The proof of Ore's conjecture has been recently completed (see [19]).…”

Section: Definably Simple Pseudofinite Groupsmentioning

confidence: 99%

Pseudofinite groups as fixed points in simple groups of finite Morley rank

Uğurlu

2013

Journal of Pure and Applied Algebra

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Section: Definably Simple Pseudofinite Groupsmentioning

confidence: 99%

Pseudofinite groups as fixed points in simple groups of finite Morley rank

Uğurlu

2013

Journal of Pure and Applied Algebra

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“…If x, y ∈ G, then [x, y] = x −1 y −1 xy is the commutator of x and y. The closed subgroup of G generated by all commutators is the commutator subgroup G ′ of G. In general, elements of G ′ need not be commutators (see for instance [11] and references therein). On the other hand, Nikolov and Segal showed that for any positive integer m there exists an integer f (m) such that if G is m-generator, then every element in G ′ is a product of at most f (m) commutators [13].…”

Section: Introductionmentioning

confidence: 99%

On profinite groups with commutators covered by nilpotent subgroups

Shumyatsky¹

2016

Rev. Mat. Iberoam.

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Abstract.The following results about a profinite group G are obtained. The commutator subgroup G ′ is finite if and only if G is covered by countably many abelian subgroups. The group G is finite-by-nilpotent if and only if G is covered by countably many nilpotent subgroups. The main result is that the commutator subgroup G ′ is finite-by-nilpotent if and only if the set of all commutators in G is covered by countably many nilpotent subgroups.

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“…The following result is classical, easy and very well-known. According to [KM07], it was first proved in [Mil99]. L. Pyber suggests using [LS01] to prove an analogous improvement on Lemma 4.9 for arbitrary finite simple groups T i .…”

Section: Babai-seress Revisitedmentioning

confidence: 99%

Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective

Helfgott¹

2019

Groups St Andrews 2017 in Birmingham

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By now, we have a product theorem in every finite simple group G of Lie type, with the strength of the bound depending only in the rank of G. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Altn, we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem.We shall revisit the proof of the bound for Altn, bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a product theorem for Altn -not of full strength, as that would be impossible, but strong enough to imply the diameter bound.

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On commutators in groups

Cited by 23 publications

References 47 publications

Pseudofinite groups as fixed points in simple groups of finite Morley rank

Pseudofinite groups as fixed points in simple groups of finite Morley rank

On profinite groups with commutators covered by nilpotent subgroups

Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective

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