Small doubling in ordered nilpotent groups of class 2

Abstract

The aim of this paper is to present a complete description of the structure of finite subsets S of a nilpotent group of class 2 satisfying |S 2 | = 3|S| − 2.

On the structure of subsets of an orderable group with some small doubling properties

“…Proposition 6 (see [11], Theorem 3.2). Let G be an orderable nilpotent group of class 2 and let S be a finite subset of G of size at least 4, such that S is nonabelian.…”

“…In papers [8], [9], [10], [11] and [12], we started the precise investigation of small doubling problems for subsets of an ordered group. We recall that if G is a group and ≤ is a total order relation defined on the set G, then (G, ≤) is an ordered group if for all a, b, x, y ∈ G the inequality a ≤ b implies that xay ≤ xby, and a group G is orderable if there exists an order ≤ on the set G such that (G, ≤) is an ordered group.…”

“…Then one of the following holds: This paper is organized as follows. In Section 2, we record some useful results from [12] and [11]. In Section 3, we first study the group a, b…”