On Kneser's addition theorem in groups

Author:
George T. Diderrich

Journal:
Proc. Amer. Math. Soc. **38** (1973), 443-451

MSC:
Primary 10L05

DOI:
https://doi.org/10.1090/S0002-9939-1973-0311617-0

MathSciNet review:
0311617

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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.

**[1]**J. H. B. Kemperman,*On complexes in a semigroup*, Nederl. Akad. Wetensch. Proc. Ser. A.**59**(1956), 247–254. MR**0085263****[2]**J. H. B. Kemperman,*On small sumsets in an abelian group*, Acta Math.**103**(1960), 63–88. MR**0110747**, https://doi.org/10.1007/BF02546525**[3]**Henry B. Mann,*Addition theorems: The addition theorems of group theory and number theory*, Interscience Publishers John Wiley & Sons New York-London-Sydney, 1965. MR**0181626**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0311617-0

Keywords:
Addition theorems,
Kneser theorem,
the Dyson -transform,
the Kemperman -transform,
commutative sets

Article copyright:
© Copyright 1973
American Mathematical Society