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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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