# Augmentation quotients for Burnside rings of some finite \(\varvec{p}\)-groups

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## Abstract

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 \(Q_n(G)\) 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.

## Keywords

Finite*p*-group Burnside ring augmentation ideal augmentation quotient

## 2010 Mathematics Subject Classification

16S34 20C05## Notes

### Acknowledgements

This work was supported by the NSFC (No. 11401155).

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© Indian Academy of Sciences 2018