An Upper Bound on the Length of an Algebra and Its Application to the Group Algebra of the Dihedral Group

Abstract

Let $\mathcal A$ be an $\mathbb F$-algebra and let $\mathcal S$ be its generating set. The length of $\mathcal S$ is the smallest number $k$ such that $\mathcal A$ equals the $\mathbb F$-linear span of all products of length at most $k$ of elements from $\mathcal S$. The length of $\mathcal A$, denoted by $l(\mathcal A)$, is defined to be the maximal length of its generating set. In this paper, it is shown that the $l(\mathcal A)$ does not exceed the maximum of $\dim \mathcal A / 2$ and $m(\mathcal A)-1$, where $m(\mathcal A)$ is the largest degree of the minimal polynomial among all elements of the algebra $\mathcal A$. For arbitrary odd $n$, it is proven that the length of the group algebra of the dihedral group of order $2n$ equals $n$.