Cyclic Extensions of Free Pro-p Groups

Herfort, Wolfgang; Zalesskiĭ, Pavel

doi:10.1006/jabr.1998.7787

Cited by 9 publications

(6 citation statements)

References 14 publications

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“…We note also that the second countability condition is essential: for bigger cardinality the result is not true, a counter example is available in [7,Section 4]. We use notation for profinite and pro-p groups from [9].…”

Section: • the Module U N Of N -Fixed Points Is A Free Z P [G/n ]-Modmentioning

confidence: 99%

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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Section: • the Module U N Of N -Fixed Points Is A Free Z P [G/n ]-Modmentioning

confidence: 99%

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

MacQuarrie

Zalesskiĭ

2016

Sel. Math. New Ser.

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“…In the proof, we use essentially Theorem 2.2 in Herfort-Zalesskii [5], which describes certain free-by-cyclic pro-p groups as a free pro-p product of normalizers of subgroups of order p and some additional free factor. If n = 1 then we use Theorem A to prove that the abelianization…”

Section: Theorem a Let C = X Be A Group Of Order P And Let M Be A Z mentioning

confidence: 99%

Cyclic extensions of free pro-$p$ groups and $p$-adic modules

Porto

Zalesskiĭ

2013

Mathematical Research Letters

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“…Apart from this not much is known about Aut(F n ), for example, it is not clear whether the virtual cohomological dimension of Aut(F n ) is finite. Certain progress however has been made in [4] and [5] towards understanding finite p-subgroups of Aut(F n ).…”

Section: Introductionmentioning

confidence: 99%

“…Let H be a cyclic group of order p r and let ρ : H → GL n (Z p ) be a representation. Then ρ lifts if and only if the Z p H-module M associated to ρ is a direct sum of indecomposable Z p H-modules which are isomorphic either to Z p C p k for some 0 ≤ k ≤ r or to J C p m (C p k ) for some 0 ≤ m < k ≤ r. The proofs of Theorem A and B are constructive in the sense that we give an explicit lifting φ : H → Aut(F n ) in terms of the semidirect product F n φ H. We use the result of [5] describing free-by-cyclic pro-p groups and a free decomposition of [4] for free-by-finite pro-p groups having finite centralizers of torsion elements. The representations in both theorems above are not necessarily faithful.…”

Section: Introductionmentioning

confidence: 99%