A Generalization of an Addition Theorem of Kneser

Abstract

AbstractA theorem of Kneser states that in an abelian group G, if A and B are finite subsets in G and AB={ab:a∈A,b∈B}, then ∣AB∣⩾∣A∣+ ∣B∣− ∣H(AB)∣ where H(AB)={g∈G:g(AB)=AB}. Motivated by the study of a problem in finite fields, we prove an analogous result for vector spaces over a field E in an extension field K of E. Our proof is algebraic and it gives an immediate proof of Kneser's Theorem