Abelian group pairs having a trivial coGalois group

Hill, Paul

doi:10.1007/s10587-008-0069-9

Cited by 1 publication

(2 citation statements)

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“…Lemma 2.2 in [10] for modules). This links work done by Enochs-Rada [10] and Hill [19] on torsion free covers of abelian groups with trivial co-Galois group, and recent work on cellular covers by Chachólski-Farjoun-Göbel-Segev [8], and Buckner-Dugas [5]. Proof.…”

Section: Envelopes and Covers With Trivial Galois Groupsmentioning

confidence: 70%

“…Our motivation to write this article was to connect notions and tools from different contexts. In particular, our aim is to connect the work by Enochs and Rada [10], as well as Hill [19] (where they considered torsion free covers of abelian groups having trivial co-Galois group) to the work by Buckner-Dugas [5] and Farjoun-Göbel-Shelah-Segev [12]. Our Theorem 3.1 ensures that, in fact, F -covers of arbitrary groups with trivial co-Galois group are cellular covers for any class F of groups.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Envelopes and Covers With Trivial Galois Groupsmentioning

confidence: 70%