Localization and finite simple groups

Parker, Christopher; Saxl, Jan

doi:10.1007/bf02771787

Cited by 5 publications

(3 citation statements)

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“…The authors also considered the concept of rigid components of non-abelian finite simple groups defined by the following equivalence relation: two such groups are related if there is a zig-zag of monomorphisms which are localizations connecting them. Parker-Saxl [22], completing results of [25], showed that all (nonabelian) finite simple groups lie in the same rigid component, except the ones of the form P Sp 4 (p 2 c ), with p an odd prime and c > 0, which are isolated. This fact motivates the following Question 4.9 Are all non-abelian finite simple groups lying in the same weak rigid component?…”

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confidence: 87%

Envelopes and covers for groups

Estrada

Rodríguez

2018

Groups Geom. Dyn.

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We connect work done by Enochs, Rada and Hill in module approximation theory with work undertaken by several group theorists and algebraic topologists in the context of homotopical localization and cellularization of spaces. This allows one to consider envelopes and covers of arbitrary groups. We show some characterizing results for certain classes of groups, and present some open questions.In a few cases it is possible to give an explicit list of localizations of a fixed group H or cellular covers of a fixed group G. For example, see the recent work by Blomgren, Chachólski, Farjoun, and Segev [3] where a complete classification of all cellular covers of each finite simple group is given.However, in many other cases the classification is not possible, as we may obtain a proper class of solutions. The use of infinite combinatorial principles, like Shelah's Black Box and its relatives, has allowed one to produce either arbitrarily large localizations or cellular covers for certain groups. For instance, in [14] and [16] the authors constructed large localizations of finite simple groups. Countable as well as arbitrarily large cellular covers of cotorsion-free abelian groups with given ranks have also been constructed (see [5], [12] [15]). On the other hand, interesting new results have been achieved in [13] where Göbel, Herden, and Shelah constructed absolute E-rings (localizations of Z) of a size below the first Erdös cardinal. This approach has yielded the solution of an old problem by Fuchs.By relaxing the uniqueness property, some of the previous constructions could be adapted to find new envelopes and covers of groups. More precisely, ϕ : H → G is an

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confidence: 87%

Envelopes and covers for groups

Estrada

Rodríguez

2018

Groups Geom. Dyn.

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“…A nC1 of alternating groups (for n 7) is a localization. Indeed, this motivated the study of localizations between simple groups in [30,32,28]. First examples of infinite localizations of A n (for n 10) were also given in [22]; in this case A n ,!…”

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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“…Libman [13] showed that the natural inclusion A n ֒→ A n+1 of alternating groups (for n ≥ 7) is a localization. This motivated the study of localizations between simple groups in [18,19,17]. First examples of infinite localizations of A n (for n ≥ 10) were also given in [13]; in this case A n ֒→ SO(n) is a localization.…”

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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Localization and finite simple groups

Cited by 5 publications

References 15 publications

Envelopes and covers for groups

Envelopes and covers for 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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