On subsets with small product in torsion-free groups. (English) Zbl 0930.20034
The main result is the following: If \(C\) is a generating subset of a nonabelian torsion-free group \(G\) with \(1\in C\) then for all \(B\subseteq G\) such that \(|B|\geq 4\), \(|BC|\geq|B|+|C|+1\). In particular, a finite subset \(X\) with cardinality \(k\geq 4\) satisfies the inequality \(|X^2|\leq 2|X|\) if and only if there are two elements \(x,r\) of \(G\), such that the conditions (i) \(xr=rx\), (ii) \(Xx=\{1,r,\dots,r^k\}\setminus\{c\}\) where \(c\in\{1,r\}\), hold.
Reviewer: Piroska Lakatos (Debrecen)
MSC:
20F05 | Generators, relations, and presentations of groups |
20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
11B75 | Other combinatorial number theory |
11B13 | Additive bases, including sumsets |
05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |