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THE BRAUER GROUP OF THE DIHEDRAL GROUP

Published online by Cambridge University Press:  19 May 2004

G. CARNOVALE
Affiliation:
Dipartimento di Matematica Pura ed Applicata, via Belzoni 7, I-35131 Padua, Italy e-mail: carnoval@math.unipd.it
J. CUADRA
Affiliation:
Universidad de Almería, Dept. Álgebra y análisis matemático, E-04120 Almería, Spain e-mail: jcdiaz@ual.es
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Abstract

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Let $p^m$ be a power of a prime number $p$, $\mathbb{Dacute;_{p^m}$ be the dihedral group of order $2p^m$ and $k$ be a field where $p$ is invertible and containing a primitive $2p^m$-th root of unity. The aim of this paper is computing the Brauer group $BM(k,\mathbb{D}_{p^m},R_z)$ of the group Hopf algebra of $\mathbb{D}_{p^m}$ with respect to the quasi-triangular structure $R_z$ arising from the group Hopf algebra of the cyclic group $\mathbb{Z}_{p^m}$ of order $p^m,$ for $z$ coprime with $p$. The main result states that $BM(k,\mathbb{D}_{p^m},R_z)\cong \mathbb{Z}_2 \times k^{\cdot}/k^{\cdot 2} \times Br(k)$ when $p$ is odd and when $p=2,$$BM(k,\mathbb{D}_{2^m},R_z) \cong \mathbb{Z}_2\times \mathbb{Z}_2 \times k^{\cdot}/k^{\cdot 2} \times k^{\cdot}/k^{\cdot 2} \times Br(k).$

Keywords

Type
Research Article
Copyright
2004 Glasgow Mathematical Journal Trust