Matchings in arbitrary groups

Eliahou, Shalom; Lecouvey, Cédric

doi:10.1016/j.aam.2007.01.002

Cited by 12 publications

(27 citation statements)

References 6 publications

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“…This latter result of Losonczy has been generalized for arbitrary groups by Eliahou and Lecouvey (see [5]). We would like to mention that, although all finite groups of prime order are known to satisfy the matching property, the classification of those prime numbers p such that Z/pZ has the acyclic matching property is unsolved.…”

Section: Introductionmentioning

confidence: 77%

On matching property for groups and field extensions

2015

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In this paper we prove a sufficient condition for the existence of matchings in arbitrary groups and its linear analogue, which lead to some generalizations of the existing results in the theory of matchings in groups and central extensions of division rings. We introduce the notion of relative matchings between arrays of elements in groups and use this notion to study the behavior of matchable sets under group homomorphisms. We also present infinite families of prime numbers p such that Z/pZ does not have the acyclic matching property. Finally, we introduce the linear version of acyclic matching property and show that purely transcendental eld extensions satisfy this property.

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Section: Introductionmentioning

confidence: 77%

On matching property for groups and field extensions

2015

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“…Our pairings are dual to pairings used in the literature [4,17,2]. The two notions are equivalent up to replacing the group by its opposite group, or replacing (A, B) by (A −1 , B −1 ).…”

Section: Introductionmentioning

confidence: 99%

“…In the abelian case, Losonczy [17] proved that if A and B are finite subsets of a prime abelian group, have the same cardinality and 1 / ∈ B, then μ(B, A) = 0. In the non-abelian case, Eliahou and Lecouvey [2] proved that if B is a finite subset of a group such that 1 / ∈ B, then μ(B, B) = 0. Two remarks on these results are worth mentioning.…”

Section: Introductionmentioning

confidence: 99%

Counting Certain Pairings in Arbitrary Groups

Hamidoune

2011

Combinator. Probab. Comp.

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In this paper, we study certain pairings which are defined as follows: if A and B are finite subsets of an arbitrary group, a Wakeford–Fan–Losonczy pairing from B onto A is a bijection φ : B → A such that bφ(b) ∉ A, for every b ∈ B. The number of such pairings is denoted by μ(B, A).We investigate the quantity μ(B, A) for A and B, two finite subsets of an arbitrary group satisfying 1 ∉ B, |A| = |B|, and the fact that the order of every element of B is ≥ |B| + 1. Extending earlier results, we show that in this case, μ(B, A) is never equal to 0. Furthermore we prove an explicit lower bound on μ(B, A) in terms of |B| and the cardinality of the group generated by B, which is valid unless A and B have a special form explicitly described. In the case A = B, our bound holds unless B is a translate of a progression.

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“…What groups possess the matching property, and when are there automatchings from B to B? The following answers were first obtained by Losonczy [12] in the abelian case, and then extended to arbitrary groups in [3]. Theorem 1.1 and 1.2 were established using methods and tools pertaining to additive number theory and combinatorics.…”

Section: Introductionmentioning

confidence: 99%

Matching subspaces in a field extension

Eliahou

Lecouvey

2010

Journal of Algebra

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In this paper, we formulate and prove linear analogues of results concerning matchings in groups. A matching in a group G is a bijection ϕ between two finite subsets A, B of G with the property, motivated by old questions on symmetric tensors, that aϕ(a) / ∈ A for all a ∈ A. Necessary and sufficient conditions on G, ensuring the existence of matchings under appropriate hypotheses, are known. Here we consider a similar question in a linear setting.Given a skew field extension K ⊂ L, where K commutative and central in L, we introduce analogous notions of matchings between finite-dimensional K -subspaces A, B of L, and obtain existence criteria similar to those in the group setting. Our tools mix additive number theory, combinatorics and algebra.

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Matchings in arbitrary groups

Cited by 12 publications

References 6 publications

On matching property for groups and field extensions

On matching property for groups and field extensions

Counting Certain Pairings in Arbitrary Groups

Matching subspaces in a field extension

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