We deduce a simple representation and the invariant factor decompositions of the subgroups of the group , where and are arbitrary positive integers. We obtain formulas for the total number of subgroups and the number of subgroups of a given order. 1. Introduction Let be the group of residue classes modulo and consider the direct product , where and are arbitrary positive integers. This paper aims to deduce a simple representation and the invariant factor decompositions of the subgroups of the group . As consequences we derive formulas for the number of certain types of subgroups of , including the total number of its subgroups and the number of its subgroups of order (). Subgroups of (sublattices of the two-dimensional integer lattice) and associated counting functions were considered by several authors in pure and applied mathematics. It is known, for example, that the number of subgroups of index in is , the sum of the (positive) divisors of . See, for example, [1, 2], [3, ]. Although features of the subgroups of not only are interesting by their own but also have applications, one of them is described below, it seems that a synthesis on subgroups of cannot be found in the literature. In the case , the subgroups of play an important role in numerical harmonic analysis, more specifically in the field of applied time-frequency analysis. Time-frequency analysis attempts to investigate function behavior via a phase space representation given by the short-time Fourier transform [4]. The short-time Fourier coefficients of a function are given by inner products with translated modulations (or time-frequency shifts) of a prototype function , assumed to be well localized in phase space, for example, a Gaussian. In applications, the phase space corresponding to discrete, finite functions (or vectors) belonging to is exactly . Concerned with the question of reconstruction from samples of short-time Fourier transforms, it has been found that when sampling on lattices, that is, subgroups of , the associated analysis and reconstruction operators are particularly rich in structure, which, in turn, can be exploited for efficient implementation (cf. [5–7] and references therein). It is of particular interest to find subgroups in a certain range of cardinality; therefore, a complete characterization of these groups helps choose the best one for the desired application. We recall that a finite Abelian group of order has rank if it is isomorphic to , where and (), which is the invariant factor decomposition of the given group. Here the number is uniquely determined and
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.
[2]
Y. M. Zou, “Gaussian binomials and the number of sublattices,” Acta Crystallographica, vol. 62, no. 5, pp. 409–410, 2006.
[3]
“The On-Line Encyclopedia of Integer Sequences,” http://oeis.org/.
[4]
K. Gr？chenig, Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkh？user, Boston, Mass, USA, 2001.
[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.
[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, Birkh？user, Boston, Mass, USA, 1998.
[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.
[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.
[12]
G. C？lug？reanu, “The total number of subgroups of a finite abelian group,” Scientiae Mathematicae Japonicae, vol. 60, no. 1, pp. 157–167, 2004.
[13]
J. Petrillo, “Counting subgroups in a direct product of finite cyclic groups,” College Mathematics Journal, vol. 42, no. 3, pp. 215–222, 2011.
[14]
M. T？rn？uceanu, “A new method of proving some classical theorems of abelian groups,” Southeast Asian Bulletin of Mathematics, vol. 31, no. 6, pp. 1191–1203, 2007.
[15]
M. T？rn？uceanu, “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.
[16]
W. C. Calhoun, “Counting the subgroups of some finite groups,” The American Mathematical Monthly, vol. 94, no. 1, pp. 54–59, 1987.
[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.
[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.
[19]
P. J. McCarthy, Introduction to Arithmetical Functions, Springer, New York, NY, USA, 1986.
[20]
W. G. Nowak and L. Tóth, “On the average number of subgroups of the group ,” International Journal of Number Theory, vol. 10, pp. 363–374, 2014.
[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.
