Schur multipliers and the Lazard correspondence

Eick, Bettina; Horn, Max; Zandi, Shiva

doi:10.1007/s00013-012-0426-7

Cited by 14 publications

(5 citation statements)

References 6 publications

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“…In order for the correspondence to apply to all the relevant objects, we need to assume that c + 1 < p. In particular, starting with a group of nilpotency class 2, this applies to all primes p > 3. Finally we obtain the natural isomorphism to obtain the same result for the case of finite p-groups, either using Schur's theory of covers (see [14]) or more explicitly via the usual interpretations of low-dimensional cohomology (see [19]).…”

Section: Transporting Homologymentioning

confidence: 86%

Irrationality of Generic Quotient Varieties via Bogomolov Multipliers

Jezernik

Sánchez

2023

International Mathematics Research Notices

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The Bogomolov multiplier of a group is the unramified Brauer group associated with the quotient variety of a faithful representation of the group. This object is an obstruction for the quotient variety to be stably rational. The purpose of this paper is to study these multipliers associated with nilpotent pro-$p$ groups by transporting them to their associated Lie algebras. Special focus is set on the case of $p$-adic Lie groups of nilpotency class $2$, where we analyse the moduli space. This is then applied to give information on asymptotic behaviour of multipliers of finite images of such groups of exponent $p$. We show that with fixed $n$ and increasing $p$, a positive proportion of these groups of order $p^n$ have trivial multipliers. On the other hand, we show that by fixing $p$ and increasing $n$, log-generic groups of order $p^n$ have non-trivial multipliers. Whence quotient varieties of faithful representations of log-generic $p$-groups are not stably rational, applications in non-commutative Iwasawa theory are developed.

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Section: Transporting Homologymentioning

confidence: 86%

Irrationality of Generic Quotient Varieties via Bogomolov Multipliers

Jezernik

Sánchez

2023

International Mathematics Research Notices

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“…For a Lie ring L, the Schur multiplier is defined as the second Chevalley-Eilenberg homology group of L. We recall the following properties of Schur multipliers of nilpotent Lie p-rings from [1].…”

Section: Preliminariesmentioning

confidence: 99%

“…The Lazard correspondence, see [5], establishes an equivalence between the categories of nilpotent Lie p-rings of class less than p and of p-groups of class less than p. This associates to any nilpotent Lie p-ring L of class less than p a p-group G(L) of the same class. The morphisms in both categories are isomorphisms and thus a classification up to isomorphism of the nilpotent Lie p-rings of order p n translates to the classification of groups of order p n for all p ≥ n. In [1] it is observed that the Schur multiplier M (L) of a nilpotent Lie p-ring of class less than p satisfies M (L) = M (G(L)). Hence the above table is also a table of the Schur multipliers of the p-groups of order dividing p 6 for all primes p ≥ 5.…”

Section: Introductionmentioning

confidence: 99%

Symbolic computation of Schur multipliers with an application to the groups of order dividing $p^6$

Eick¹,

Ghorbanzadeh²

2018

Preprint

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“…First we give a brief summary of the Lazard correspondence (see [17,11]) between the category of nilpotent Lie rings L of nilpotency class c and order p n , p > c, and the category G of finite p-group of nilpotency class c and order p n . For each L ∈ L we denote by Gr(L) ∈ G the group with the same set of elements and with multiplication defined by the Beiker-Campbell-Hausdorff formula (BCH-formula, [17]), which has the form…”

Section: A Construction Of a P-group By A Graphmentioning

confidence: 99%

On Borel complexity of the isomorphism problems for graph related classes of Lie algebras and finite p-groups

Lipyanski

Vanetik

2015

J. Algebra Appl.

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Communicated by E. ZelmanovWe reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for some class of finite-dimensional 2-step nilpotent Lie algebras over a field and for some class of finite p-groups. We show that the isomorphism problem for graphs is harder than the two latter isomorphism problems in the sense of Borel reducibility. A computable analogue of Borel reducibility was introduced by S. Coskey, J. D. Hamkins, and R. Miller (2012). A relation of the isomorphism problem for undirected graphs to the well-known problem of classifying pairs of matrices over a field (up to similarity) is also studied.

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Schur multipliers and the Lazard correspondence

Cited by 14 publications

References 6 publications

Irrationality of Generic Quotient Varieties via Bogomolov Multipliers

Irrationality of Generic Quotient Varieties via Bogomolov Multipliers

Symbolic computation of Schur multipliers with an application to the groups of order dividing $p^6$

On Borel complexity of the isomorphism problems for graph related classes of Lie algebras and finite p-groups

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