On finite intersections of ?henselian valued? fields

Heinemann, Bernhard

doi:10.1007/bf01171485

Cited by 13 publications

(6 citation statements)

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“…Note that v i and v i are independent by Remark 5.10 and Lemma 5.11(5). We recall now the following special case of a result from [Heinemann 1985]:…”

Section: Proof By Theorem 54 the Isomorphism G(kmentioning

confidence: 99%

Realizing profinite reduced special groups

Astier¹,

Mariano²

2011

Pacific J. Math.

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“…Note that v i and v i are independent by Remark 5.10 and Lemma 5.11(5). We recall now the following special case of a result from [Heinemann 1985]:…”

Section: Proof By Theorem 54 the Isomorphism G(kmentioning

confidence: 99%

Realizing profinite reduced special groups

Astier¹,

Mariano²

2011

Pacific J. Math.

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“…This answers in the negative a conjecture of Ershov [2, p. 426]. (This is also implicit in a theorem of Heinemann [11,Theorem 3.2]: if I is a prime and K1 and K2 are the fixed fields of the I-Sylow subgroups of G(Q p ), G(Q q ), respectively, then there are a, T E G(Q) such that G(Kf n KD is a pro-I-group. In particular,…”

Section: Relative Projective Groupsmentioning

confidence: 52%

On Closed Subgroups of Free Products of Profinite Groups

Haran

1987

Proceedings of the London Mathematical Society

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IntroductionIf G is a free product of a family {Ai}ieI of discrete groups then a subgroup H of G is the free product of a free group F and (Af n H), where a E ~(i), i E I, and ~(i) is a set of representatives of Ai \ G / H. This is the content of the Kurosh subgroup theorem (KST). Is a similar result true for closed subgroups of free (profinite) products of pro finite groups? (Say, with F projective instead of free.)An answer to this question requires an appropriate definition of a free product over an infinite family of groups. Such a definition has been proposed, by Gildenhuys and Ribes in [3], for groups indexed by compact topological spaces so that the factors are locally equal to each other, except for neighbourhoods of one distinguished point. In spite of the fact that the KST holds for open subgroups of such free products, this definition seems to be too restrictive: if H is a closed subgroup of the free product then the groups Af n H, with a E G, i E I, need not be 'locally equal' to each other (cf. Example 2.4).We propose a very natural generalization of the free product with finitely many factors: an inverse limit of such free products (over an inverse system with mappings that send respective factors again into factors of a free product). This, essentially, also includes the definition of [3].We do not know whether the analogue of the KST holds for open subgroups of these free products. Nevertheless, if we restrict ourselves to separable groups, we give a satisfactory account of the closed subgroups of the free products.1. The analogue of the KST does not hold, in general, for closed subgroups of free products (Example 5.5).2. We define for a pro finite group G the notion of projectivity relative to a given family I of its subgroups (Definition 4.2). We show:2a. if G is a free product of the groups in I, and H is a closed subgroup of G, then H is projective relative to {fa n HI f E I, a E G};2b. conversely, if H is separable and projective relative to ~ then H is a closed subgroup of a free product G of a family I of subgroups such that ~ = {fa n HI rEI, a E G}.3. Separable relative projective pro-p-groups are in fact free products (Corollary 9.6).Hence we can answer a question of Lubotzky [13, 2.10]: 4. The KST holds for separable closed subgroups of free pro-p-products.

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“…4.3] in the case where n = 2 and whereK 1 ,K 2 are p-henselizations of K with respect to non-trivial valuations with residue characteristic / = p (however, there is an inaccuracy in the proof in p. 708, line 13). Parts (b) and (c) are pro-p analogues of the results of [He,§1]. Proposition 4.3.…”

Section: 3]mentioning

confidence: 87%

Free pro- $p$ product decompositions of Galois groups

Efrat

1997

Math Z

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On finite intersections of ?henselian valued? fields

Cited by 13 publications

References 5 publications

Realizing profinite reduced special groups

Realizing profinite reduced special groups

On Closed Subgroups of Free Products of Profinite Groups

Free pro- $p$ product decompositions of Galois groups

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