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

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.

References in corpus (3)

• A Survey of Graph Pebbling
• Recent Progress in Graph Pebbling
• An application of graph pebbling to zero-sum sequences in abelian groups