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Astérisque - Titles - 1999 - 258 - pages 281-308

Titles1999258

Structure Theory of Set Addition
Jean-Marc Deshouillers, Bernard Landreau, Alexander A. Yudin (Ed.)
Astérisque 258 (1999), 458 pages
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On small subset product in a group
Yahya Ould Hamidoune
Astérisque 258 (1999), 281-308

Résumé :
Nous généralisons des théorèmes d'addition connus pour le cas des groupes non abéliens.

Les preuves classiques des théorèmes d'addition utilisent des transformations locales dues à Davenport, Dyson et Kempermann.

Notre approche est basée sur l'étude de certains blocs d'imprimitivité du groupe d'automorphismes d'une relation.

Abstract:
We generalise some known addition theorems to non abelian groups and to the most general case of relations having a transitive group of automorphisms.

The classical proofs of addition theorems use local transformations due to Davenport, Dyson and Kempermann. We present a completely different method based on the study of some blocks of imprimitivity with respect to the automorphism group of a relation.

Several addition theorems including the finite $\alpha + \beta $-Theorem of Mann and a formula proved by Davenport and Lewis will be generalised to relations having a transitive group of automorphisms.

We study the critical pair theory in the case of finite groups. We generalise Vosper Theorem to finite not necessarily abelian groups.

Chowla, Mann and Straus obtained in 1959 a lower bound for the size of the image of a diagonal form on a prime field. This result was generalised by Tietäväienen to finite fields with odd characteristics. We use our results on the critical pair theory to generalise this lower bound to an arbitrary division ring.

Our results apply to the superconnectivity problems in networks. In particular we show that a loopless Cayley graph with optimal connectivity has only trivial minimum cuts when the degree and the order are coprime.

Key words: Addition theorems, blocks of imprimitivity, network reliability.

Class. math. : Primary: 20D60, Secondary: 20K01, 11B13, 11B75, 05C25.


ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique