Articles written in Proceedings – Mathematical Sciences

    • Augmentation quotients for real representation rings of cyclic groups


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      Denote by $C_{m}$ the cyclic group of order $m$. Let $\mathcal{R}(C_{m})$ be its real representation ring, and $\Delta(C_{m})$ its augmentation ideal. In this paper, we give an explicit $\mathbb{Z}$-basis for the $n$-th power $\Delta^{n}(C_{m})$ and determine the isomorphism class of the $n$-th augmentation quotient $\Delta^{n}(C_{m})/\Delta^{n+1}(C_{m})$ for each positive integer $n$.

    • Augmentation quotients for Burnside rings of some finite $p$-groups


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      Let $G$ be a finite group, $\Omega(G)$ be its Burnside ring and $\Delta(G)$ the augmentation ideal of $\Omega(G)$. Denote by $\Delta^{n}(G)$ and $\mathcal{Q}_{n}(G)$ the $n$-th power of $\Delta(G)$ and the $n$-th consecutive quotient group $\Delta^{n}(G)/\Delta^{n+1}(G)$, respectively. This paper provides an explicit $\mathbb{Z}$-basis for $\Delta^{n}(\mathcal{H})$ and determine the isomorphism class of $\mathcal{Q}_{n}(\mathcal{H})$ for each positive integer $n$, where $\mathcal{H} = \langle g, h| g^{p^{m}} = h^{p} = 1, h^{−1}gh = g^{p^{m−1}+1}\rangle$, $p$ is an odd prime

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