Finite p-groups all of whose d-generated subgroups are powerful

Snopce, Ilir

doi:10.1016/j.jalgebra.2014.09.014

Cited by 1 publication

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“…Observe that the "if" part of the theorem holds in greater generality (Proposition 3.7); we also remark that the condition p > d makes it possible to apply Lie methods (see Section 3, in particular, Remark 3.2). It is worth noting that during the last decade the groups listed in Theorem A have appeared in the literature in different contexts (see, for example, [12,13,[20][21][22]24]). The reader will find a more detailed account of the related results at the end of Section 3.…”

Section: Introductionmentioning

confidence: 99%

On hereditarily self-similar $p$-adic analytic pro-$p$ groups

Noseda

Snopce

2022

Groups Geom. Dyn.

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A non-trivial finitely generated pro-p group G is said to be strongly hereditarily selfsimilar of index p if every non-trivial finitely generated closed subgroup of G admits a faithful selfsimilar action on a p-ary tree. We classify the solvable torsion-free p-adic analytic pro-p groups of dimension less than p that are strongly hereditarily self-similar of index p. Moreover, we show that a solvable torsion-free p-adic analytic pro-p group of dimension less than p is strongly hereditarily self-similar of index p if and only if it is isomorphic to the maximal pro-p Galois group of some field that contains a primitive pth root of unity. As a key step for the proof of the above results, we classify the 3-dimensional solvable torsion-free p-adic analytic pro-p groups that admit a faithful self-similar action on a p-ary tree, completing the classification of the 3-dimensional torsion-free p-adic analytic pro-p groups that admit such actions.Dedicated to Said Sidki on the occasion of his 80th birthday.

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