On Kneser's addition theorem in groups
Abstract: The following theorem is proved.
Theorem A. Let be a group written additively with finite nonempty subsets . Assume that is commutative, i.e. , for . Then there exists an Abelian subgroup of such that and .
This is Kneser's theorem, if is Abelian. Also, as an application of the above theorem, the following is proved.
Theorem B. Let be a finite group of order and let be a sequence (repeats are allowed) of nonzero elements of . The set of sums where and must contain a nontrivial subgroup of .
Finally, the Kemperman -transform, a transform similar to the Dyson -transform, is introduced and evidence is given to support the conjecture that Theorem A remains true, if the commutativity of is dropped.
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Keywords: Addition theorems, Kneser theorem, the Dyson -transform, the Kemperman -transform, commutative sets
Article copyright: © Copyright 1973 American Mathematical Society