Product-one subsequences over subgroups of a finite group
Volume 189 / 2019
Acta Arithmetica 189 (2019), 209-221
MSC: Primary 20D60; Secondary 11B75.
DOI: 10.4064/aa180103-22-8
Published online: 23 May 2019
Abstract
Let $G$ be a finite group, and let $\mathsf D^{(1)}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ with length $|S|\geq t$ has a nonempty subsequence $T$ over a cyclic subgroup of $G$ with the product of all terms in $T$ in some order equal to one, the identity element of $G$. We prove that $\mathsf D^{(1)}(G)\geq |G|$ for all finite groups $G$ and characterize all nilpotent finite groups such that equality holds. When $G$ is abelian, we also provide a computation formula for $\mathsf D^{(1)}(G)$ involving the Möbius function.
