Rigidity of p-adic p-Torsion

Weiss, Alfred

doi:10.2307/2007056

Cited by 91 publications

(99 citation statements)

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“…Theorem 2.7 is inspired by Theorem 2 of the article [13] by A. Weiss and our proof follows his closely. The theorems in this section are of independent interest.…”

Section: Modulesmentioning

confidence: 97%

Second countable virtually free pro-p groups whose torsion elements have finite centralizer

MacQuarrie

Zalesskiĭ

2016

Sel. Math. New Ser.

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A second countable virtually free pro-p group all of whose torsion elements have finite centralizer is the free pro-p product of finite p-groups and a free pro-p factor. The proof explores a connection between p-adic representations of finite p-groups and virtually free pro-p groups. In order to utilize this connection, we first prove a version of a remarkable theorem of A. Weiss for infinitely generated profinite modules that allows us to detect freeness of profinite modules. The proof now proceeds using techniques developed in the combinatorial theory of profinite groups. Using an HNN-extension we embed our group into a semidirect product F ⋊ K of a free pro-p group F and a finite p-group K that preserves the conditions on centralizers and such that every torsion element is conjugate to an element of K. We then show that the ZpK-module F/[F, F ] is free using the detection theorem mentioned above. This allows us to deduce the result for F ⋊ K, and hence for our original group, using the pro-p version of the Kurosh subgroup theorem.

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“…Theorem 2.7 is inspired by Theorem 2 of the article [13] by A. Weiss and our proof follows his closely. The theorems in this section are of independent interest.…”

Section: Modulesmentioning

confidence: 97%

Second countable virtually free pro-p groups whose torsion elements have finite centralizer

MacQuarrie

Zalesskiĭ

2016

Sel. Math. New Ser.

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“…This conjecture was confirmed for Nilpotent Groups in [2,3] and cyclic-by-abelian groups in [4]. However, these techniques do not transfer to simple groups.…”

mentioning

confidence: 88%

“…Let G = 3 D 4 (2 3 ). Clearly |G| = 211341312 = 2 12 · 3 4 · 7 2 · 13 and ex p(G) = 6552 = 2 3 · 3 2 · 7 · 13.…”

Section: Proof Of Theoremunclassified

“…In order to prove that the Zassenhaus Conjecture holds, we need to consider torsion units of V (ZG) of order 2,3,4,6,7,8,9,12,13,14,18,21,28,24,26,36,39,42,56,63 and 91 (by Proposition 1). For the purpose of this paper and due to the complexity of certain orders, we shall consider elements of order 2, 3, 7, 13, 26, 39 and 91.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…G contains elements of order 6, 14 and 21. Therefore [2,3], [2,7] and [3,7] are adjacent in π(G) and consequently adjacent in π(V (ZG)). Clearly π(G) = π(V (ZG)), since there are no torsion units of order 26, 39 and 91 in V (ZG).…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Torsion units for a Ree group, Tits group and a Steinberg triality group

Gildea

2015

Rend. Circ. Mat. Palermo

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Abelianization and fixed point properties of units in integral group rings

Bächle

Janssens

Jespers

et al. 2022

Mathematische Nachrichten

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Let 𝐺 be a finite group and  (ℤ𝐺) the unit group of the integral group ring ℤ𝐺. We prove a unit theorem, namely, a characterization of when  (ℤ𝐺) satisfies Kazhdan's property (T), both in terms of the finite group 𝐺 and in terms of the simple components of the semisimple algebra ℚ𝐺. Furthermore, it is shown that for  (ℤ𝐺), this property is equivalent to the weaker property FAb (i.e., every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property FA, denoted HFA. More precisely, it is described when all subgroups of finite index in  (ℤ𝐺) have both finite abelianization and are not a nontrivial amalgamated product. A crucial step for this is a reduction to arithmetic groups SL 𝑛 (), where  is an order in a finite-dimensional semisimple ℚ-algebra 𝐷, and finite groups 𝐺, which have the so-called cut property. For such groups 𝐺, we describe the simple epimorphic images of ℚ𝐺. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups E 𝑛 (𝐷) of SL 𝑛 (𝐷). These groups are well understood except in the degenerate case of lower rank, that is, for SL 2 () with  an order in a division algebra 𝐷 with a finite number of units. In this setting, we determine Serre's property FA for E 2 () and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its ℤ-rank.

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Rigidity of p-adic p-Torsion

Cited by 91 publications

References 4 publications

Second countable virtually free pro-p groups whose torsion elements have finite centralizer

Second countable virtually free pro-p groups whose torsion elements have finite centralizer

Torsion units for a Ree group, Tits group and a Steinberg triality group

Abelianization and fixed point properties of units in integral group rings

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