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

Let $(G,1_G)$ be a finite group and let $S=g_1\text{bdot}\dots \text{bdot}{g}_{\mathrm{\ell }}$ 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 ${\sum }_{i=1}^{\mathrm{\ell }}\frac{1}{\text{ord}(g_i)}\le 1$. Let $\mathsf{t}\mathsf{i}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ with $\mathrm{\mid }S\mathrm{\mid }\ge t$ has a tiny product-one subsequence. The direct problem is to obtain the exact value of $\mathsf{t}\mathsf{i}(G)$, 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.