Revised version for publication in Israel J. Math. EXTERIOR ALGEBRAS AND TWO CONJECTURES ON FINITE ABELIAN GROUPS

Let G be a finite abelian group with |G| > 1. Let a1, . . . , ak be k distinct elements of G and let b1, . . . , bk be (not necessarily distinct) elements of G, where k is a positive integer smaller than the smallest prime divisor of |G|. We show that there is a permutation π on {1, . . . , k} such that a1bπ(1), . . . , akbπ(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.