Augmentation quotients for Burnside rings of generalized dihedral groups

Chang, Shan

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1165-1175

Résumé

Let $H$ be a finite abelian group of odd order, $\mathcal {D}$ be its generalized dihedral group, i.e., the semidirect product of $C_2$ acting on $H$ by inverting elements, where $C_2$ is the cyclic group of order two. Let $\Omega (\mathcal {D})$ be the Burnside ring of $\mathcal {D}$, $\Delta (\mathcal {D})$ be the augmentation ideal of $\Omega (\mathcal {D})$. Denote by $\Delta ^n(\mathcal {D})$ and $Q_n(\mathcal {D})$ the $n$th power of $\Delta (\mathcal {D})$ and the $n$th consecutive quotient group $\Delta ^n(\mathcal {D})/\Delta ^{n+1}(\mathcal {D})$, respectively. This paper provides an explicit $\mathbb {Z}$-basis for $\Delta ^n(\mathcal {D})$ and determines the isomorphism class of $Q_n(\mathcal {D})$ for each positive integer $n$.

Détail

DOI : 10.1007/s10587-016-0316-4

Classification : 16S34, 20C05
Keywords: generalized dihedral group; Burnside ring; augmentation ideal; augmentation quotient