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ISSN 1088-6826(online) ISSN 0002-9939(print)



On Kneser's addition theorem in groups

Author: George T. Diderrich
Journal: Proc. Amer. Math. Soc. 38 (1973), 443-451
MSC: Primary 10L05
MathSciNet review: 0311617
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Abstract: The following theorem is proved.

Theorem A. Let $ G$ be a group written additively with finite nonempty subsets $ A,B$. Assume that $ B$ is commutative, i.e. $ {b_1} + {b_2} = {b_2} + {b_1}$, for $ {b_1},{b_2} \in B$. Then there exists an Abelian subgroup $ H$ of $ G$ such that $ A + B + H = A + H + B = A + B$ and $ \vert A + B\vert \geqq \vert A + H\vert + \vert B + H\vert - \vert H\vert$.

This is Kneser's theorem, if $ G$ is Abelian. Also, as an application of the above theorem, the following is proved.

Theorem B. Let $ G$ be a finite group of order $ \upsilon (\upsilon > 1)$ and let $ {a_1}, \cdots ,{a_\upsilon }$ be a sequence (repeats are allowed) of nonzero elements of $ G$. The set $ S$ of sums $ {a_{{i_1}}} + \cdots + {a_{{i_t}}}$ where $ 1 \leqq {i_1} < \cdots < {i_t} \leqq \upsilon $ and $ 1 \leqq t \leqq \upsilon $ must contain a nontrivial subgroup $ H$ of $ G$.

Finally, the Kemperman $ d$-transform, a transform similar to the Dyson $ e$-transform, is introduced and evidence is given to support the conjecture that Theorem A remains true, if the commutativity of $ B$ is dropped.

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Keywords: Addition theorems, Kneser theorem, the Dyson $ e$-transform, the Kemperman $ d$-transform, commutative sets
Article copyright: © Copyright 1973 American Mathematical Society

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