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

TL;DR

This paper explicitly describes the structure of augmentation quotients of Burnside rings for a specific class of finite $p$-groups, providing bases and isomorphism classifications for these algebraic objects.

Contribution

It offers an explicit $bZ$-basis for the powers of the augmentation ideal and classifies the associated quotient groups for a particular finite $p$-group.

Findings

Explicit $bZ$-basis for $ riangle^n(bH)$

Determination of isomorphism classes of $Q_n(bH)$

Structural insights into Burnside ring quotients for the group $bH$

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 $\mathcal{H}=\langle g,h |\, g^{p^m}=h^p=1, h^{-1}gh=g^{p^{m-1}+1}\rangle$, $p$ is an odd prime.