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Keywords:
generalized dihedral group; Burnside ring; augmentation ideal; augmentation quotient
generalized dihedral group; Burnside ring; augmentation ideal; augmentation quotient
Summary:
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 $n$.
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