Cauchy–davenport Theorem in Group Extensions
نویسنده
چکیده
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 has a representation in the form ab, where a ∈ A and b ∈ 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.
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