Augmentation Quotients for Burnside Rings of some Finite $p$-Groups

Abstract

Let $G$ be a finite group, $\Omega (G)$ be its Burnside ring, and $\Delta (G)$ its augmentation ideal. Denote by ${\Delta }^n(G)$ and $Q_n(G)$ 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 $Q_n(\mathcal{H})$ for each positive integer $n$, where $\mathcal{H}=⟨g,h|g^{p^m}=h^p=1,h^{-1}gh=g^{p^{m-1}+1}⟩$, $p$ is an odd prime.