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Deltahedron (talk | contribs) →References: Geroldinger et al (2009) |
Deltahedron (talk | contribs) cite Geroldinger & Rusza (2009) |
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In mathematics, in the field of [[additive combinatorics]], '''Kneser's theorem''', named after [[Martin Kneser]], is a statement about [[Sumset|set addition]] in [[finite group]]s.<ref>{{cite journal | first=Martin | last=Kneser | title=Abschätzungen der asymptotischen Dichte von Summenmengen | language=German | journal=[[Math. Zeitschr.]] | volume=58 | year=1953 | pages=459–484 | zbl=0051.28104 }}
</ref> It may be regarded as an extension of the [[Cauchy-Davenport theorem]] on sumsets in groups of prime order.<ref name=GR143>Geroldinger & Rusza (2009) p.143</ref>
==Statement==
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:<math>\begin{align} |A+B| &\ge |A+H| + |B+H| - |H| \\ &\ge |A| + |B| - |H|. \end{align} </math>
The subgroup ''H'' can be taken to be the ''stabiliser''<ref name=GR143/> of ''A''+''B''
:<math> H = \lbrace g \in G : g + (A+B) = (A+B) \rbrace. </math>
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