AN UPPER BOUND ON THE LENGTH OF AN ALGEBRA AND ITS APPLICATION TO THE GROUP ALGEBRA OF THE DIHEDRAL GROUP
M. A. Khrystik · Bulletin of the Australian Mathematical Society · 2025
Abstract Let ${\mathcal {A}}$ be a unital ${\mathbb {F}}$ -algebra and let ${\mathcal {S}}$ be a generating set of ${\mathcal {A}}$ . 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 sets. We show that $l({\mathcal {A}})$ 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 ${\mathcal {A}}$ . As an application, we show that for arbitrary odd n, the length of the group algebra of the dihedral group of order $2n$ equals n.