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AN UPPER BOUND ON THE LENGTH OF AN ALGEBRA AND ITS APPLICATION TO THE GROUP ALGEBRA OF THE DIHEDRAL GROUP
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 10 February 2025, pp. 1-10
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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.
