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

  • SHAN CHANGEmail author
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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).

References

  1. 1.
    Bak A and Tang Guoping, Solutions to the presentation problem for powers of the augmentation ideal of torsion free and torsion abelian groups, Adv. Math. 189 (2004) 1–37Google Scholar
  2. 2.
    Chang Shan and Tang Guoping, A basis for augmentation quotients of finite abelian groups, J. Algebra 327 (2011) 466–488MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chang Shan, Chen Hong and Tang Guoping, Augmentation quotients for complex representation rings of dihedral groups, Front. Math. China 7 (2012) 1–18MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chang S, Augmentation quotients for complex representation rings of point groups, J. Anhui Univ. Nat. Sci. 38 (2014) 13–19MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chang Shan, Augmentation quotients for complex representation rings of generalized quaternion groups, Chin. Ann. Math. Ser. B 37 (2016) 571–584MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chang Shan, Augmentation quotients for Burnside rings of generalized dihedral groups, Czech. Math. J. 66(4) (2016) 1165–1175MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chang S and Liu H, Augmentation quotients for real representation rings of cyclic groups, Proc. Indian Acad. Sci. (Math. Sci.) 128 (2018) 48,  https://doi.org/10.1007/s12044-018-0415-2
  8. 8.
    Magurn BA (2002) An algebraic introduction to \(K\)-theory. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  9. 9.
    Parmenter M M, A basis for powers of the augmentation ideal, Algebra Colloq. 8 (2001) 121–128MathSciNetzbMATHGoogle Scholar
  10. 10.
    Tang Gaohua, Li Yu and Wu Yansheng, On the consecutive quotients for Burnside ring of some nonabelian 2-groups, J. Guangxi Teach. Edu. Univ. Nat. Sci. 33(2) (2016) 1–7Google Scholar
  11. 11.
    Tang Guoping, Presenting powers of augmentation ideals of elementary p-groups. K-Theory, 23 (2001) 31-39MathSciNetCrossRefGoogle Scholar
  12. 12.
    Tang Guoping, On a problem of Karpilovsky, Algebra Colloq. 10 (2003) 11–16MathSciNetCrossRefGoogle Scholar
  13. 13.
    Tang Guoping, Structure of augmentation quotients of finite homocyclic abelian groups, Sci. China Ser. A. 50 (2007) 1280–1288MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wen Y and Chang S, Augmentation quotients for Burnside rings of some finite groups of order \(p^3\), College Math. 33 (2017) 113–117Google Scholar
  15. 15.
    Wu H and Tang G, The structure of powers of the augmentation ideal and their consecutive quotients for the Burnside ring of a finite abelian group. Adv. Math. China 36 (2007) 627–630MathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.School of MathematicsHefei University of TechnologyHefeiChina

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