Hamidoune, Y. O. An isoperimetric method in additive theory. (English) Zbl 0842.20029 J. Algebra 179, No. 2, 622-630 (1996). 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) Cited in 1 ReviewCited in 20 Documents MSC: 20E34 General structure theorems for groups 20D60 Arithmetic and combinatorial problems involving abstract finite groups 05C75 Structural characterization of families of graphs Keywords:\(k\)-atoms; \(k\)-critical sets; \(k\)-isoperimetric numbers; reflexive graphs; subsets in groups; critical pairs of subsets PDFBibTeX XMLCite \textit{Y. O. Hamidoune}, J. Algebra 179, No. 2, 622--630 (1996; Zbl 0842.20029) Full Text: DOI