Harborth Constants for Certain Classes of Metacyclic Groups

TL;DR

This paper calculates the Harborth constants for specific metacyclic groups, extending previous results on dihedral groups, and characterizes subsets that do not contain the required product of elements.

Contribution

It generalizes Harborth constant computations to a new class of metacyclic groups and solves the inverse problem for these groups.

Findings

Harborth constants for $H_{n,m}$ are explicitly determined.

Characterization of subsets lacking the product condition.

Extension of known results from dihedral to metacyclic groups.

Abstract

The Harborth constant of a finite group $G$ is the smallest integer $k\geq \exp(G)$ 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 $H_{n, m}=\langle x, y \mid x^n=1, y^2=x^m, yx=x^{-1}y \rangle$. We also solve the "inverse" problem of characterizing all smaller subsets that do not contain $\exp(H_{n,m})$ distinct elements whose product is $1$.