The Harborth Constant of Dihedral Groups

Abstract

The Harborth constant of a finite group $G$, denoted $\gs(G)$, is the smallest integer $k$ such that the following holds: For $A\subseteq G$ with $|A|=k$, there exists $B\subseteq A$ with $|B|=\exp(G)$ such that the elements of $B$ can be rearranged into a sequence whose product equals $1_G$, the identity element of $G$. The Harborth constant is a well studied combinatorial invariant in the case of abelian groups. In this paper, we consider a generalization $\gs(G)$ of this combinatorial invariant for nonabelian groups and prove that if $G$ is a dihedral group of order $2n$ with $n\ge 3$, then $\gs(G) = n + 2$ if $n$ is even and $\gs(G) = 2n + 1$ otherwise.