On some subgroup chains related to Kneser’s theorem

Yahya Ould Hamidoune[1]; Oriol Serra[2]; Gilles Zémor[3]

  • [1] Université Pierre et Marie Curie, Paris 6 Combinatoire et Optimisation - case 189 4 place Jussieu 75252 Paris Cedex 05, France
  • [2] Universitat Politècnica de Catalunya Matemàtica Aplicada IV Campus Nord - Edif. C3 C. Jordi Girona, 1-3 08034 Barcelona, Spain.
  • [3] Institut de Mathématiques de Bordeaux Université de Bordeaux 1 351 cours de la Libération 33405 Talence, France.

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 1, page 125-130
  • ISSN: 1246-7405

Abstract

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A recent result of Balandraud shows that for every subset S of an abelian group G there exists a non trivial subgroup H such that | T S | ≤ | T | + | S | - 2 holds only if H ⊂ S t a b ( T S ) . Notice that Kneser’s Theorem only gives { 1 } ≠ S t a b ( T S ) .This strong form of Kneser’s theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabelian groups and, by using classical tools from Additive Number Theory, extend some of the above results. In particular we obtain short proofs of Balandraud’s results in the abelian case.

How to cite

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Hamidoune, Yahya Ould, Serra, Oriol, and Zémor, Gilles. "On some subgroup chains related to Kneser’s theorem." Journal de Théorie des Nombres de Bordeaux 20.1 (2008): 125-130. <http://eudml.org/doc/10824>.

@article{Hamidoune2008,
abstract = {A recent result of Balandraud shows that for every subset $S$ of an abelian group $G$ there exists a non trivial subgroup $H$ such that $|TS|\le |T|+|S|-2$ holds only if $H\subset Stab (TS)$. Notice that Kneser’s Theorem only gives $\lbrace 1\rbrace \ne Stab (TS)$.This strong form of Kneser’s theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabelian groups and, by using classical tools from Additive Number Theory, extend some of the above results. In particular we obtain short proofs of Balandraud’s results in the abelian case.},
affiliation = {Université Pierre et Marie Curie, Paris 6 Combinatoire et Optimisation - case 189 4 place Jussieu 75252 Paris Cedex 05, France; Universitat Politècnica de Catalunya Matemàtica Aplicada IV Campus Nord - Edif. C3 C. Jordi Girona, 1-3 08034 Barcelona, Spain.; Institut de Mathématiques de Bordeaux Université de Bordeaux 1 351 cours de la Libération 33405 Talence, France.},
author = {Hamidoune, Yahya Ould, Serra, Oriol, Zémor, Gilles},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Kneser's theorem; non-abelian groups; additive number theory},
language = {eng},
number = {1},
pages = {125-130},
publisher = {Université Bordeaux 1},
title = {On some subgroup chains related to Kneser’s theorem},
url = {http://eudml.org/doc/10824},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Hamidoune, Yahya Ould
AU - Serra, Oriol
AU - Zémor, Gilles
TI - On some subgroup chains related to Kneser’s theorem
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 1
SP - 125
EP - 130
AB - A recent result of Balandraud shows that for every subset $S$ of an abelian group $G$ there exists a non trivial subgroup $H$ such that $|TS|\le |T|+|S|-2$ holds only if $H\subset Stab (TS)$. Notice that Kneser’s Theorem only gives $\lbrace 1\rbrace \ne Stab (TS)$.This strong form of Kneser’s theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabelian groups and, by using classical tools from Additive Number Theory, extend some of the above results. In particular we obtain short proofs of Balandraud’s results in the abelian case.
LA - eng
KW - Kneser's theorem; non-abelian groups; additive number theory
UR - http://eudml.org/doc/10824
ER -

References

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  1. E. Balandraud, Une variante de la méthode isopérimétrique de Hamidoune, appliquée au théorème de Kneser. Annales de l’institut Fourier, to appear. 
  2. E. Balandraud, Quelques résultats combinatoires en théorie additive des nombres. Thèse de doctorat de l’Université de Bordeaux I, May 2006. 
  3. D. Grynkiewicz, A step beyond Kemperman’s structure Theorem. Preprint Oct. 2007. Zbl1213.11179
  4. J. H. B. Kemperman, On small sumsets in Abelian groups. Acta Math. 103 (1960), 66–88. Zbl0108.25704MR110747
  5. M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen. Math. Zeit. 58 (1953), 459–484. Zbl0051.28104MR56632
  6. M. Kneser, Summenmengen in lokalkompakten abelesche Gruppen. Math. Zeit. 66 (1956), 88–110. Zbl0073.01702MR81438
  7. R. A. Lee, Proving Kneser’s theorem for finite groups by another e -transform. Proc. Amer. Math. Soc. 44 (1974), 255–258. Zbl0316.20023
  8. H. B. Mann, Addition Theorems. R.E. Krieger, New York, 1976. Zbl0189.29701MR424744
  9. M. B. Nathanson, Additive Number Theory. Inverse problems and the geometry of sumsets. Grad. Texts in Math. 165, Springer, 1996. Zbl0859.11003MR1477155
  10. J. E. Olson, On the symmetric difference of two sets in a group. European J. Combin. 7 (1986), 43–54. Zbl0597.05012MR850143
  11. T. Tao, V. H. Vu, Additive Combinatorics. Cambridge Studies in Advanced Mathematics 105, Cambridge University Press, 2006. Zbl1127.11002MR2289012

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