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An isoperimetric method in additive theory. (English) Zbl 0842.20029
This paper deals with subsets in groups \(G\) showing that certain conditions above all on their cardinality control their structure. The main theorem reads as follows: Let \(G=\langle B\rangle\), with finite \(B\) containing \(1\). If every element of \(G\setminus\{1\}\) has order \(\geq |B|\), then either (1) \(|XB|\geq\min(|G|-1,|X|+|B|)\) for all finite, at least 2-element subsets \(X\); or (2) there are \(r\neq 1\) and \(j\in\mathbb{Z}\) such that \(B=\{r^i;\) \(j\leq i\leq j+|B|-1\}\). Reformulated as a theorem on critical pairs of subsets it includes earlier results in the literature, among others of A. G. Vosper for prime \(|G|\).
Reviewer: G.Kowol (Wien)

MSC:
20E34 General structure theorems for groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
05C75 Structural characterization of families of graphs
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