Group varieties not closed under cellular covers and localizations

Herden, Daniel; Petapirak, Montakarn; Rodríguez, J.L.

doi:10.1515/jgth-2017-0021

Cited by 3 publications

(3 citation statements)

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“…Moreover, mutatis mutandis, they may also be used to describe the Z/p-acyclic cover of these Burnside groups of high exponent. This is related with recent results about cellular covers of Burnside groups ( [25], [28] [22], [23]), that in particular have been useful to solve some conjectures by Farjoun [17]. On the other hand, note that we have approached the description of the acyclic and cellular covers of a non-nilpotent group with respect to an uncountable non-nilpotent group.…”

Section: The Burnside Radical and A Free Burnside Groupsupporting

confidence: 61%

Generators and closed classes of groups

Flores¹,

Rodríguez²

2021

Publicacions Matemàtiques

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We show that, in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits, and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. In this way, we extend to the group-theoretic framework the topological analogue proved by Chachólski, Parent, and Stanley in 2004. We also establish several new relations between singly-generated closed classes.

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Section: The Burnside Radical and A Free Burnside Groupsupporting

confidence: 61%

Generators and closed classes of groups

Flores¹,

Rodríguez²

2021

Publicacions Matemàtiques

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“…More related to our approach in this paper, (co)localizations of Burnside groups (which in turn are themselves localizations of free groups) have appeared at least in two different contexts: in relation with amenability phenomena [13], and from the point of view of combinatorial group theory, for example in [18] and [19]. In particular, the existence of 2 @ 0 varieties of groups not closed under cellular covers (another name for colocalizations) is shown in the last references, complementing the fact that there are 2 @ 0 varieties closed under taking cellular covers [14].…”

Section: Introductionmentioning

confidence: 99%

On localizations of quasi-simple groups with given countable center

Flores

Rodríguez

2020

Groups Geom. Dyn.

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A group homomorphism iW H ! G is a localization of H , if for every homomorphism 'W H ! G there exists a unique endomorphism W G ! G such that i D ' (maps are acting on the right). Göbel and Trlifaj asked in [18, Problem 30.4(4), p. 831] which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e., a central extension of a simple group. The proof uses Obraztsov and Ol 0 shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Thévenaz and Viruel.

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“…In this paper we apply Obraztsov [15] to find localizations of (quasi)-simple groups with given countable center, see Theorem 2.3 These ideas come from [11], where the existence of 2 ℵ 0 varieties of groups not closed under cellular covers is shown; this complements the fact that there are 2 ℵ 0 varieties closed under taking cellular covers [6]. Recall that cellular covers of simple groups were described in [1], see also [3].…”

Section: Introductionmentioning

confidence: 99%

On localizations of quasi-simple groups with given countable center

Flores¹,

Rodríguez²

2018

Preprint

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A group homomorphism i : H → G is a localization of H, if for every homomorphism ϕ : H → G there exists a unique endomorphism ψ : G → G, such that iψ = ϕ (maps are acting on the right). Göbel and Trlifaj asked in [10, Problem 30.4(4), p. 831] which abelian groups are centers of localizations of simple groups.Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e. a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Thévenaz and Viruel.

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Group varieties not closed under cellular covers and localizations

Abstract: A group homomorphism

Cited by 3 publications

References 20 publications

Generators and closed classes of groups

Generators and closed classes of groups

On localizations of quasi-simple groups with given countable center

On localizations of quasi-simple groups with given countable center

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