Exterior algebras and two conjectures on finite abelian groups

TL;DR

This paper proves a conjecture about permutations in finite abelian groups using exterior algebras, confirms a special case for p-groups, and proposes a new conjecture linking determinants and characters.

Contribution

It introduces a novel approach using exterior algebras to confirm a conjecture on element distinctness in abelian groups and proposes a new conjecture related to determinants and characters.

Findings

Confirmed the Dasgupta-Karolyi-Serra-Szegedy conjecture for abelian p-groups.

Established conditions for permutations ensuring element distinctness in finite abelian groups.

Proposed a new conjecture involving determinants and characters, implying Snevily's conjecture for odd order groups.

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 $\pi$ on {1,...,k} such that a_1b_{\pi(1)},...,a_kb_{\pi(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-Karolyi-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.