Abstract
Let (G,1G) be a finite group and let S=g1⋅…⋅gℓ 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 1G and ∑i=1ℓ1ord(gi)≤1. Let ti(G) be the smallest integer t such that every sequence S over G with |S|≥t has a tiny product-one subsequence. The direct problem is to obtain the exact value of 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.
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