Table of Contents
Journal of Numbers
Volume 2014, Article ID 491428, 6 pages
http://dx.doi.org/10.1155/2014/491428
Research Article

Representing and Counting the Subgroups of the Group

1NuHAG, Faculty of Mathematics, University of Vienna, Oskar Morgenstern Platz 1, 1090 Vienna, Austria
2Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040 Vienna, Austria
3Department of Mathematics, University of Pécs, Ifjúság Útja 6, Pécs 7624, Hungary
4Institute of Mathematics, University of Natural Resources and Life Sciences, Gregor-Mendel-Straße 33, 1180 Vienna, Austria

Received 28 July 2014; Accepted 14 September 2014; Published 16 October 2014

Academic Editor: Andrej Dujella

Copyright © 2014 Mario Hampejs et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. J. Grady, “A group theoretic approach to a famous partition formula,” The American Mathematical Monthly, vol. 112, no. 7, pp. 645–651, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. Y. M. Zou, “Gaussian binomials and the number of sublattices,” Acta Crystallographica, vol. 62, no. 5, pp. 409–410, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. “The On-Line Encyclopedia of Integer Sequences,” http://oeis.org/.
  4. K. Gröchenig, Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, Mass, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  5. G. Kutyniok and T. Strohmer, “Wilson bases for general time-frequency lattices,” SIAM Journal on Mathematical Analysis, vol. 37, no. 3, pp. 685–711 (electronic), 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. A. J. van Leest, Non-separable Gabor schemes: their design and implementation [Ph.D. thesis], Technische Universiteit Eindhoven, Eindhoven, The Netherlands, 2001.
  7. T. Strohmer, “Numerical algorithms for discrete Gabor expansions,” in Gabor Analysis and Algorithms: Theory and Applications, H. G. Feichtinger and T. Strohmer, Eds., pp. 267–294, Birkhäuser, Boston, Mass, USA, 1998. View at Google Scholar
  8. A. Machì, Groups: An Introduction to Ideas and Methods of the Theory of Groups, Springer, 2012.
  9. J. J. Rotman, An Introduction to the Theory of Groups, vol. 148 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 4th edition, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  10. R. Schmidt, Subgroup Lattices of Groups, de Gruyter Expositions in Mathematics 14, de Gruyter, Berlin, Germany, 1994.
  11. M. Suzuki, “On the lattice of subgroups of finite groups,” Transactions of the American Mathematical Society, vol. 70, pp. 345–371, 1951. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. Călugăreanu, “The total number of subgroups of a finite abelian group,” Scientiae Mathematicae Japonicae, vol. 60, no. 1, pp. 157–167, 2004. View at Google Scholar · View at MathSciNet
  13. J. Petrillo, “Counting subgroups in a direct product of finite cyclic groups,” College Mathematics Journal, vol. 42, no. 3, pp. 215–222, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. M. Tărnăuceanu, “A new method of proving some classical theorems of abelian groups,” Southeast Asian Bulletin of Mathematics, vol. 31, no. 6, pp. 1191–1203, 2007. View at Google Scholar · View at MathSciNet
  15. M. Tărnăuceanu, “An arithmetic method of counting the subgroups of a finite abelian group,” Bulletin Mathématiques de la Société des Sciences Mathématiques de Roumanie, vol. 53, no. 101, pp. 373–386, 2010. View at Google Scholar
  16. W. C. Calhoun, “Counting the subgroups of some finite groups,” The American Mathematical Monthly, vol. 94, no. 1, pp. 54–59, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  17. L. Tóth, “Menon's identity and arithmetical sums representing functions of several variables,” Rendiconti del Seminario Matematico Università e Politecnico di Torino, vol. 69, no. 1, pp. 97–110, 2011. View at Google Scholar · View at MathSciNet · View at Scopus
  18. L. Tóth, “On the number of cyclic subgroups of a finite Abelian group,” Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie, vol. 55, no. 103, pp. 423–428, 2012. View at Google Scholar
  19. P. J. McCarthy, Introduction to Arithmetical Functions, Springer, New York, NY, USA, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  20. W. G. Nowak and L. Tóth, “On the average number of subgroups of the group Zm × Zn,” International Journal of Number Theory, vol. 10, pp. 363–374, 2014. View at Publisher · View at Google Scholar
  21. K. Bauer, D. Sen, and P. Zvengrowski, “A generalized Goursat lemma,” http://arxiv.org/abs/1109.0024.
  22. A. Pakapongpun and T. Ward, “Functorial orbit counting,” Journal of Integer Sequences, vol. 12, Article ID 09.2.4, 20 pages, 2009. View at Google Scholar · View at MathSciNet