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Journal of Mathematical Chemistry

, Volume 53, Issue 7, pp 1537–1548 | Cite as

Zeta and normal zeta functions for a subclass of space groups

  • Hermina Alajbegović
  • Muharem Avdispahić
Original Paper

Abstract

We calculate zeta and normal zeta functions of space groups with the point group isomorphic to the cyclic group of order 2. The obtained results are applied to determine the number of subgroups, resp. normal subgroups, of a given index for each of these groups.

Keywords

Zeta functions of groups Subgroup growth Crystallographic groups 

Mathematics Subject Classification

11M41 20H15 

References

  1. 1.
    M.P.F. du Sautoy, A. Lubotzky, Functional equations and uniformity for local zeta functions of nilpotent groups. Am. J. Math. 118(4), 39–90 (1996)CrossRefGoogle Scholar
  2. 2.
    M.P.F. du Sautoy, J.J. McDermott, G.C. Smith, Zeta functions of crystallographic groups and analytic continuation. Proc. Lond. Math. Soc. 79, 511–534 (2010)CrossRefGoogle Scholar
  3. 3.
    M.P.F. du Sautoy, L. Woodward, in Zeta Functions of Groups and Rings. Lecture Notes in Mathematics 1925, (Springer, 2008)Google Scholar
  4. 4.
    F.J. Grunewald, D. Segal, G.C. Smith, Subgroups of finite index in nilpotent groups. Invent. Math. 93, 185–223 (1988)CrossRefGoogle Scholar
  5. 5.
    E. Lord, Generators and relations for space groups. Bangalore (2010)Google Scholar
  6. 6.
    J.J. McDermott, Subgroup growth and the plane crystallographic groups. Ph.D., University of Bath, UK (1997)Google Scholar
  7. 7.
    G.C. Smith, Zeta-functions of torsion-free finitely generated nilpotent groups. Ph.D., University of Manchester, UK (1983)Google Scholar
  8. 8.
    C. Voll, Functional equations for zeta functions of groups and rings. Ann. Math. 172, 1181–1218 (2010)CrossRefGoogle Scholar
  9. 9.
    C. Voll, A. Beauville, Functional equations for local normal zeta functions of nilpotent groups. Geom. Funct. Anal. GAFA 15, 274–295 (2005)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Faculty of Mechanical EngineeringUniversity of ZenicaZenicaBosnia and Herzegovina
  2. 2.Department of MathematicsUniversity of SarajevoSarajevoBosnia and Herzegovina

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