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Cauchy-Davenport theorem in group extensions. (English) Zbl 1111.20026

Summary: Let \(A\) and \(B\) be nonempty subsets of a finite group \(G\) in which the order of the smallest nonzero subgroup is not smaller than \(d=|A|+|B|-1\). Then at least \(d\) different elements of \(G\) have a representation in the form \(ab\), where \(a\in A\) and \(b\in B\). This extends a classical theorem of Cauchy and Davenport to noncommutative groups. We also generalize Vosper’s inverse theorem in the same spirit, giving a complete description of critical pairs \(A,B\) for which exactly \(d\) group elements can be written in the form \(ab\). The proofs depend on the structure of group extensions.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
11B75 Other combinatorial number theory
11P70 Inverse problems of additive number theory, including sumsets
20E22 Extensions, wreath products, and other compositions of groups
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