Augmentation quotients for Burnside rings of generalized dihedral groups

Czechoslovak Mathematical Journal

Shan Chang¹ 

[1] School of Mathematics, Hefei University of Technology, No. 193, Tunxi Road, Hefei 230009, Anhui, People's Republic of China

关键词: generalized dihedral group;    Burnside ring;    augmentation ideal;    augmentation quotient;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

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 $n$.

【 授权许可】

Unknown