%0 Journal Article
%T Representing and Counting the Subgroups of the Group
%A Mario Hampejs
%A Nicki Holighaus
%A L¨¢szl¨® T¨®th
%A Christoph Wiesmeyr
%J Journal of Numbers
%D 2014
%R 10.1155/2014/491428
%X 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¨C7] 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
%U http://www.hindawi.com/journals/jn/2014/491428/