Harborth Constants for Certain Classes of Metacyclic Groups

Abstract

The Harborth constant of a finite group $G$ is the smallest integer $k\ge \exp (G)$ such that any subset of $G$ of size $k$ contains $\exp (G)$ distinct elements whose product is $1$. Generalizing previous work on the Harborth constants of dihedral groups, we compute the Harborth constants for the metacyclic groups of the form $H_{n,m}=⟨x,y\mid x^n=1,y^2=x^m,yx=x^{-1}y⟩$. We also solve the "inverse" problem of characterizing all smaller subsets that do not contain $\exp (H_{n,m})$ distinct elements whose product is $1$.