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Let G be a finite abelian group with |G|>1. Let a_1,...,a_k be k distinctelements 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|. Weshow that there is a permutation $\pi$ on {1,...,k} such thata_1b_{\pi(1)},...,a_kb_{\pi(k)} are distinct, provided that any other primedivisor of |G| (if there is any) is greater than k!. This in particularconfirms the Dasgupta-Karolyi-Serra-Szegedy conjecture for abelian p-groups. Wealso pose a new conjecture involving determinants and characters, and show thatits validity implies Snevily's conjecture for abelian groups of odd order. Ourmethods involve exterior algebras and characters.

Exterior algebras and two conjectures on finite abelian groups