A CHARACTERIZATION OF UNIFORM PRO-p GROUPS

Klopsch, Benjamin; Snopce, Ilir

doi:10.1093/qmath/hau005

Cited by 9 publications

(10 citation statements)

References 15 publications

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“…As a consequence we can provide a positive answer to the Question 1.9 in [10] for p ≥ 5. (1) G is powerful,…”

Section: The Characterization Of Powerful P-groups and Uniform Pro-p mentioning

confidence: 93%

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A characterization of powerful p-groups

González-Sánchez

Zugadi-Reizabal

2014

Isr. J. Math.

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“…As a consequence we can provide a positive answer to the Question 1.9 in [10] for p ≥ 5. (1) G is powerful,…”

Section: The Characterization Of Powerful P-groups and Uniform Pro-p mentioning

confidence: 93%

“…In [10] Klopsch and Snopce have conjectured that for an odd prime p a torsion-free pro-p group is uniform if and only if the minimal number of generators and the dimension coincide. They proved this conjecture in the special case when the group is solvable leaving the general case open.…”

Section: Introductionmentioning

confidence: 99%

A characterization of powerful p-groups

González-Sánchez

Zugadi-Reizabal

2014

Isr. J. Math.

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“…Note that in this case G/K(G) is a herediteraly uniform pro-p group (cf. [17] and [24]); in particular, if θ is the trivial homomorphism, then G/K(G) is a free abelian pro-p group. set of the group.…”

Section: Smoothness Conjecturementioning

confidence: 99%

Right-angled Artin pro-$p$ groups

Snopce¹,

Zalesskiĭ²

2020

Preprint

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Let p be a prime. The right-angled Artin pro-p group G Γ associated to a fnite simplicial graph Γ is the pro-p completion of the right-angled Artin group associated to Γ. We prove that the following assertions are equivalent: (i) no induced subgraph of Γ is a square or a line with four vertices (a path of length 3); (ii) every closed subgroup of G Γ is itself a right-angled Artin pro-p group (possibly infinitely generated); (iii) G Γ is a Bloch-Kato pro-p group; (iv) every closed subgroup of G Γ has torsion free abelianization; (v) G Γ occurs as the maximal pro-p Galois group G K (p) of some field K containing a primitive pth root of unity; (vi) G Γ can be constructed from Z p by iterating two group theoretic operations, namely, direct products with Z p and free pro-p products. This settles in the affirmative a conjecture of Quadrelli and Weigel. Also, we show that the Smoothness Conjecture of De Clercq and Florens holds for rightangled Artin pro-p groups. Moreover, we prove that G Γ is coherent if and only if each circuit of Γ of length greater than three has a chord.

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“…The hereditarily uniform pro-p groups were classified in [19]. It turns out that a uniform pro-p group G is hereditarily uniform if and only if it has a constant generating number on open subgroups, that is, d(U) = d(G) for every open subgroup U of G (cf.…”

Section: Proposition 22 Every Maximal Abelian Subgroup Of a Frattini-...mentioning

confidence: 99%

Frattini-injectivity and Maximal pro-$p$ Galois groups

Snopce¹,

Tanushevski²

2020

Preprint

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We call a pro-p group G Frattini-injective if distinct finitely generated subgroups of G have distinct Frattinis. This paper is an initial effort toward a systematic study of Frattini-injective pro-p groups (and several other related concepts). Most notably, we classify the p-adic analytic and the solvable Frattini-injective pro-p groups, and we describe the lattice of normal abelian subgroups of a Frattini-injective pro-p group.We prove that every maximal pro-p Galois group of a field that contains a primitive pth root of unity (and also contains √ −1 if p = 2) is Frattiniinjective. In addition, we show that many substantial results on maximal pro-p Galois groups are in fact consequences of Frattini-injectivity. For instance, a p-adic analytic or solvable pro-p group is Frattini-injective if and only if it can be realized as a maximal pro-p Galois group of a field that contains a primitive pth root of unity (and also contains √ −1 if p = 2); and every Frattini-injective pro-p group contains a unique maximal abelian normal subgroup.

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A CHARACTERIZATION OF UNIFORM PRO-p GROUPS

Cited by 9 publications

References 15 publications

A characterization of powerful p-groups

A characterization of powerful p-groups

Right-angled Artin pro-$p$ groups

Frattini-injectivity and Maximal pro-$p$ Galois groups

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