Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T13:16:29.527Z Has data issue: false hasContentIssue false
  • English
  • Français

SMALL DOUBLING IN ORDERED GROUPS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  30 April 2014

GREGORY FREIMAN ,
MARCEL HERZOG ,
PATRIZIA LONGOBARDI  and
MERCEDE MAJ
GREGORY FREIMAN
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel email [email protected]
MARCEL HERZOG*
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel email [email protected]
PATRIZIA LONGOBARDI
Affiliation:
Dipartimento di Matematica e Informatica, Università di Salerno, via Ponte don Melillo, 84084 Fisciano (Salerno), Italy email [email protected]
MERCEDE MAJ
Affiliation:
Dipartimento di Matematica e Informatica, Università di Salerno, via Ponte don Melillo, 84084 Fisciano (Salerno), Italy email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ is a finite subset of an ordered group that generates a nonabelian ordered group, then $|S^2|\geq 3|S|-2$. This generalizes a classical result from the theory of set addition.

Keywords

finite subsetsnilpotent groupsordered groupssmall doubling

MSC classification

Secondary: 20F60: Ordered groups 20F18: Nilpotent groups 20F99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 96 , Issue 3 , June 2014 , pp. 316 - 325
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Bilu, Y., ‘Structure of sets with small sumsets’, Asterique 258 (1999), 77–108.Google Scholar
Botto Mura, R. and Rhemtulla, A., Orderable Groups, Lecture Notes in Pure and Applied Mathematics (Marcel Dekker Inc., New York, Basel, 1977).Google Scholar
Chang, M., ‘A polynomial bound in Freiman’s theorem’, Duke Math. J. 113 (2002), 399–419.Google Scholar
Freiman, G. A., Foundations of a Structural Theory of Set Addition, Translations of Mathematical Monographs, 37 (American Mathematical Society, Providence, Rhode Island, 1973).Google Scholar
Glass, A. M. W., Partially Ordered Groups, Series in Algebra, 7 (World Scientific Publishing Co., Inc., River Edge, NJ, 1999).Google Scholar
Green, B. J. and Ruzsa, I. Z., ‘Freiman’s theorem in an arbitrary abelian group’, J. Lond. Math. Soc. (2) 75 (2007), 163–175.Google Scholar
Green, B. J. and Tao, T. C., ‘Freiman’s theorem in finite fields via extremal set theory’, Combin. Probab. Comput. 18 (2009), 335–355.Google Scholar
Hamidoune, Y. O., Llado, A. S. and Serra, O., ‘On subsets with small product in torsion-free groups’, Combinatorica 18 (1998), 529–540.Google Scholar
Iwasawa, K., ‘On linearly ordered groups’, J. Math. Soc. Japan 1 (1948), 1–9.Google Scholar
Kargapolov, M. I., ‘Completely orderable groups’, Algebra Logica 1 (1962), 16–21.Google Scholar
Levi, F. W., ‘Arithmetische Gesetze im Gebiete diskreter Gruppen’, Rend. Circ. Mat. Palermo (2) 35 (1913), 225–236.Google Scholar
Mal’cev, A. I., ‘On ordered groups’, Izv. Akad. Nauk. SSSR Ser. Mat. 13 (1948), 473–482.Google Scholar
Neumann, B. H., ‘On ordered groups’, Amer. J. Math. 71 (1949), 1–18.Google Scholar
Ruzsa, I. Z., ‘Generalized arithmetic progressions and sumsets’, Acta Math. Hungar. 65 (1994), 379–388.Google Scholar
Sanders, T., ‘A note on Freiman’s theorem in vector spaces’, Combin. Probab. Comput. 17 (2008), 297–305.CrossRefGoogle Scholar
Tao, T. C., ‘Product set estimates for non-commutative groups’, Combinatorica 28 (2008), 547–594.Google Scholar