Exterior algebras and two conjectures on finite abelian groups |
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Authors: | Tao Feng Zhi-Wei Sun Qing Xiang |
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Affiliation: | 1.Department of Mathematical Sciences,University of Delaware,Newark,USA;2.Department of Mathematics,Nanjing University,Nanjing,People’s Republic of China |
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Abstract: | Let G be a finite abelian group with |G| > 1. Let a 1, …, a k be k distinct elements of G and let b 1, …, b k be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that there is a permutation π on {1, …,k} such that a 1 b π(1), …, a k b π(k) are distinct, provided that any other prime divisor of |G| (if there is any) is greater than k!. This in particular confirms the Dasgupta-Károlyi-Serra-Szegedy conjecture for abelian p-groups. We also pose a new conjecture involving determinants and characters, and show that its validity implies Snevily’s conjecture for abelian groups of odd order. Our methods involve exterior algebras and characters. |
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