Let $(G,1_G)$ be a finite group and let $S=g_1\mathbf{\cdot }\dots \mathbf{\cdot }{g}_{\ell }$ 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 ${\sum }_{i=1}^{\ell }\frac{1}{\mathrm{ord}(g_i)}\le 1$. 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 subsequence. 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.

让 $（G，{\mathrm{1个}}_G）$ 成为一个有限的群体，让 $\mathrm{小号}=G_{\mathrm{1个}}\mathbf{\cdot }\dots \mathbf{\cdot }{G}_{\ell }$是G上的非空序列。我们说S是一个很小的乘积序列，如果可以对它们的项进行排序以使它们的乘积等于${\mathrm{1个}}_G$ 和 ${\sum }_{\mathrm{一世}=\mathrm{1个}}^{\ell }\frac{\mathrm{1个}}{\mathrm{奥德}（G_{\mathrm{一世}}）}\le \mathrm{1个}$。让$\mathsf{ti}（G）$是最小的整数吨，使得每个序列š超过ģ与$|\mathrm{小号}|\ge Ť$有一个很小的乘积一子序列 直接的问题是要获得$\mathsf{ti}（G）$，而反问题是表征G上没有微小乘积一子序列的长序列的结构。在本文中，我们考虑了环状群的逆问题，同时研究了二面体群和二环群的正反问题。