Harborth constants for certain classes of metacyclic groups

Abstract

The Harborth constant of a finite group G is the smallest integer k ≥ 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 Hn,m = x, y | 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(Hn,m) distinct elements whose product is 1.2010 Mathematics Subject Classification. 11B30 (primary) and 05D05 (secondary).Unfortunately, the proof in [3] is incorrect, and rectifying the errors is nontrivial. In this paper, we build on the techniques developed by Balachandran, Mazumdar, and Zhao in order to prove more general results about metacyclic groups, and the Harborth constants of dihedral groups are a special case of our results.1.2. Notation. The general finite metacyclic group has the presentation H n,p,m,r = x, y | x n = 1, y p = x m , yx = x −r y with order |H n,p,m,r | = np (where n, p ≥ 2 and m, r ≥ 0). In this paper, we focus on the classes of metacyclic groups where p = 2 and r = 1, in which case we write H n,m = x, y | x n = 1, y 2 = x m , yx = x −1 y .