Normal Subgroups of Free Profinite Groups

We study the probability of generating a finite simple group, together with its generalisation P G,soc G (d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ soc G. We prove that P G,soc G (2) 53/90, with equality if and only if G is A 6 or S 6 , and establish a similar result for P G,soc G (3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel nikov on profinite groups, as well as to a conjecture of Holt and Stather, follow easily from our results.

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“…In [9], Fireman shows that proving this conjecture would in turn resolve a 1978 problem of Mel nikov [36]. We complete this work.…”

Section: Introductionmentioning

confidence: 73%

The probability of generating a finite simple group

2013

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“…For each cardinal number m there exists a unique (up to an isomorphism) free pro-Σ-groupF m (Σ) of rank m. This group has a subset X of cardinality m which converges to 1 such that each continuous map ϕ 0 of X into a pro-Σ group G uniquely extends to a homomorphism ϕ :F m (Σ) → G. By Melnikov [Mel,Lemma 2.2],F m (Σ) has the embedding property [FrJ,p. 353 …”

Section: σ-Groupsmentioning

confidence: 99%

On Σ-Hilbertian fields

Fried¹,

Jarden²

1998

Pacific J. Math.

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“…As n i ≥ 5 these groups A ni are simple. Hence Γ ∼ = A mj for some subsequence (m j ) of (n i ) (see [M;Lemma 1.3]; in fact Γ is the product of countably many copies of ∞ n=5 A n ). The fixed field of Γ in K is Hilbertian since it is an abelian extension of the Hilbertian field k [FrJ;Thm.…”

Section: Corollary 1: Suppose K Is a Countable Hilbertian Field Of Chmentioning

confidence: 99%

The Embedding Problem Over a Hilbertian PAC-Field

Fried¹,

Völklein²

1992

The Annals of Mathematics

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Abstract:We show that the absolute Galois group of a countable Hilbertian P(seudo)-A(lgebraically)C(losed) field of characteristic 0 is a free profinite group of countably infinite rank (Theorem A). As a consequence, G(Q/Q) is the extension of groups with a fairly simple structure (e.g., ∞ n=2 S n ) by a countably free group. In addition, we characterize those PAC fields over which every finite group is a Galois group as those with the RG-Hilbertian property (Theorem B).

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Normal Subgroups of Free Profinite Groups

Cited by 28 publications

References 6 publications

The probability of generating a finite simple group

The probability of generating a finite simple group

On Σ-Hilbertian fields

The Embedding Problem Over a Hilbertian PAC-Field

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