On an inverse problem of Erd\H os, Kleitman, and Lemke

TL;DR

This paper investigates the inverse problem related to tiny product-one sequences in finite groups, focusing on cyclic, dihedral, and dicyclic groups, and aims to characterize sequences lacking such subsequences.

Contribution

It provides new characterizations of long sequences without tiny product-one subsequences for specific finite groups, advancing understanding of their structural properties.

Findings

Characterized the structure of sequences over cyclic groups without tiny product-one subsequences.

Extended analysis to dihedral and dicyclic groups for both direct and inverse problems.

Determined the minimal length thresholds for the existence of tiny product-one subsequences.

Abstract

Let $(G, 1_G)$ be a finite group and let $S=g_1\bdot \ldots\bdot g_{\ell}$ be a nonempty sequence over $G$. We say $S$ is a tiny product-one sequence if its terms can be ordered such that their product equals $1_G$ and $\sum_{i=1}^{\ell}\frac{1}{\ord(g_i)}\le 1$. Let $\mathsf {ti}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ with $|S|\ge t$ has a tiny product-one subsequence. The direct problem is to obtain the exact value of $\mathsf {ti}(G)$, while the inverse problem is to characterize the structure of long sequences over $G$ which have no tiny product-one subsequences. In this paper, we consider the inverse problem for cyclic groups and we also study both direct and inverse problems for dihedral groups and dicyclic groups.