Exterior algebras and two conjectures on finite abelian groups - Mathematics > Group Theory

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Abstract: 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 {\pi1},

.,a kb {\pik} 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.

Autor: Tao Feng, Zhi-Wei Sun, Qing Xiang

Fuente: https://arxiv.org/