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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 grisha@post.tau.ac.il
MARCEL HERZOG*
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel email herzogm@post.tau.ac.il
PATRIZIA LONGOBARDI
Affiliation:
Dipartimento di Matematica e Informatica, Università di Salerno, via Ponte don Melillo, 84084 Fisciano (Salerno), Italy email plongobardi@unisa.it
MERCEDE MAJ
Affiliation:
Dipartimento di Matematica e Informatica, Università di Salerno, via Ponte don Melillo, 84084 Fisciano (Salerno), Italy email mmaj@unisa.it
*
✉herzogm@post.tau.ac.il
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Abstract

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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. 

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