Abstract
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 represents the minimal number of generators of the group. For general accounts on finite Abelian groups see, for example, [8, 9].
It is known that for every finite Abelian group, the problem of counting all subgroups and the subgroups of a given order reduces to -groups, which follows from the properties of the subgroup lattice of the group (see [10, 11]). In particular, for , this can be formulated as follows. Assume that . Then is an Abelian group of rank two, since , where and . Let and be the prime power factorizations of and , respectively, where (). Then where and with some exponents ().
Now consider the -group , where . This is of rank two for . One has the simple explicit formulae:
Formula (3) was derived by Călugăreanu [12, Section 4] and recently by Petrillo [13, Proposition 2] using Goursat’s lemma for groups. Tărnăuceanu [14, Proposition 2.9] and [15, Theorem 3.3] deduced (3) and (4) by a method based on properties of certain attached matrices.
Therefore, and can be computed using (1), (3) and (2), (4), respectively. We deduce other formulas for and (Theorems 3 and 4), which generalize (3) and (4) and put them in more compact forms. These are consequences of a simple representation of the subgroups of , given in Theorem 1. This representation might be known, but the only source we could find is the paper [5], where only a special case is treated in a different form. More exactly, in [5, Lemma 4.1] a representation for lattices in of redundancy 2, that is, subgroups of having index , is given, using matrices in Hermite normal form. Theorem 2 gives the invariant factor decompositions of the subgroups of . We also consider the number of cyclic subgroups of (Theorem 5) and the number of subgroups of a given exponent in (Theorem 8).
Our approach is elementary, using only simple group-theoretic and number-theoretic arguments. The proofs are given in Section 4.
Throughout the paper we use the following notations: , , and are the number and the sum, respectively, of the positive divisors of , is the Dedekind function, stands for the number of distinct prime factors of , is the Möbius function, denotes Euler’s totient function, and is the Riemann zeta function.
2. Subgroups of
The subgroups of can be identified and visualized in the plane with sublattices of the lattice . Every two-dimensional sublattice is generated by two basis vectors. For example, Figure 1 shows the subgroup of having the basis vectors and .
This suggests the following representation of the subgroups.
Theorem 1. For every , let
and, for , define
Then is a subgroup of order of and the map is a bijection between the set and the set of subgroups of .
Note that for the subgroup , the basis vectors mentioned above are and , where
This notation for will be used also in the rest of the paper. Note also that in the case , the area of the parallelogram spanned by the basis vectors is , exactly the index of .
We say that a subgroup is a subproduct of if , where and are subgroups of and , respectively.
Theorem 2. (i) The invariant factor decomposition of the subgroup is given by
where
satisfying .
(ii) The exponent of the subgroup is .
(iii) The subgroup is cyclic if and only if .
(iv) The subgroup is a subproduct if and only if and . Here is cyclic if and only if .
For example, for the subgroup represented by Figure 1 one has , , , , , and , and this subgroup is isomorphic to . It is not cyclic and is not a subproduct.
According to Theorem 1, the number of subgroups of can be obtained by counting the elements of the set . We deduce the following.
Theorem 3. For every , is given by
Formula (10) is a special case of a formula representing the number of all subgroups of a class of groups formed as cyclic extensions of cyclic groups, deduced by Calhoun [16] and having a laborious proof. Note that formula (10) is given, without proof in [3, ].
Note also that the function is representing a multiplicative arithmetic function of two variables; that is, holds for any such that . This property, which is in concordance with (1), is a direct consequence of formula (10). See Section 5.
Let denote the number of solutions of the system of equations , , .
Theorem 4. For every , such that ,
The identities (3) and (4) can be easily deduced from each of the identities given in Theorems 3 and 4, respectively.
Theorem 5. Let .
(i) The number of cyclic subgroups of is given by
(ii) The number of subproducts of is and the number of its cyclic subproducts is .
Formula (17), as a special case of an identity valid for arbitrary finite Abelian groups, was derived by the third author [17, 18] using different arguments. The function is also multiplicative.
3. Subgroups of
In the case , which is of special interest in applications, the results given in the previous section can be easily used. We point out that and are multiplicative arithmetic functions of a single variable (sequences [3, ]). They can be written in the form of Dirichlet convolutions as shown by the next corollaries.
Corollary 6. For every ,
Corollary 7. For every ,
Further convolutional representations can also be given; for example, all of these follow from the Dirichlet-series representations valid for , .
Observe that which is a simple consequence of (23) or of (24) and (25). It also follows from the next result.
Theorem 8. For every with , the number of subgroups of exponent of equals the number of cyclic subgroups of .
4. Proofs
Proof of Theorem 1. Let be a subgroup of . Consider the natural projection given by . Then is a subgroup of and there is a unique divisor of such that . Let be minimal such that .
Furthermore, consider the natural inclusion given by . Then is a subgroup of and there exists a unique divisor of such that .
We show that . Indeed, for every , . On the other hand, for every one has and hence there is such that . We obtain , and there is with .
Here a necessary condition is that (obtained for and ), that is, , equivalent to . Clearly, if this is verified, then for the above representation of it is enough to take the values and .
Also, dividing by we have with and , showing that , by its minimality. Hence with . Thus we obtain the given representation.
Conversely, every generates a subgroup of order of and the proof is complete.
Proof of Theorem 2. (i)-(ii) We first determine the exponent of the subgroup . is generated by and ; hence, its exponent is the least common multiple of the orders of these two elements. The order of is . To compute the order of note that if and only if . Thus the order of is . We deduce that the exponent of is
For every finite Abelian group the rank of a nontrivial subgroup is at most the rank of the group. Therefore, the rank of is or . That is, with certain such that . Here the exponent of equals that of , which is . Using (ii) already proved we deduce that . Since the order of is we have .
(iii) According to (i), where . Hence is cyclic if and only if .
(iv) The subgroups of are of form , where , and the properties follow from (6) and (iii).
Proof of Theorem 3. By its definition, the number of elements of the set is
representing . This is formula (10).
To obtain formula (11) apply the Gauss formula () by writing the following:
Now (12) follows from (11) by the Busche-Ramanujan identity (cf. [19, Chapter 1]):
Proof of Theorem 4. According to Theorem 1, giving (13), which can be written, by Gauss’ formula again, as where in the inner sum one has and obtains (14). Now, to get (15) write (32) as and the proof is complete.
Proof of Theorem 5. (i) According to Theorems 1 and 2/(iii) and using that or and according to or ,
where the inner sum is . Hence
Now regrouping the terms according to and we obtain
which is (16).
The next results follow applying Gauss’ formula and the Busche-Ramanujan formula, similar to the proof of Theorem 3.
(ii) For the subproducts the values and can be chosen arbitrary and it follows at once that the number of subproducts is . The number of cyclic subproducts is
by the inverse Busche-Ramanujan identity.
Proof of Theorem 8. According to Theorem 2/(ii), the number of subgroups of exponent of is
Write , , and with . We deduce, similar to the proof of Theorem 5/(i), that
which is exactly (cf. (35)).
5. Further Remarks
(1)As mentioned in Section 2 the functions and are multiplicative functions of two variables. This follows easily from formulae (10) and (17), respectively. Namely, according to those formulae and are two variables Dirichlet convolutions of the functions and , respectively, with the constant 1 function, all multiplicative. Since convolution preserves the multiplicativity we deduce that and are also multiplicative. See [17, Section 2] for details.(2)Asymptotic formulas with sharp error terms for the sums and were given in the paper [20].(3)For any finite groups and a subgroup of is cyclic if and only if and have coprime orders, where and are the natural inclusions ([21, Theorem 4.2]). In the case , , and one has and and the characterization of the cyclic subgroups given in Theorem 2/(iii) can be obtained also in this way. It turns out that regarding the sublattice, is cyclic if and only if the numbers of points on the horizontal and vertical axes, respectively, are relatively prime. Note that in the case the above condition reads . Thus it is necessary that . The subgroup on Figure 1 is not cyclic.(4)Note also the next formula for the number of cyclic subgroups of , derived in [22, Example 2]: where the sum is over all ordered pairs such that . For a short direct proof of (40) write , with . Then , , and obtain according to (21).(5)Every subgroup of has the representation , where and are unique integers. This follows like in the proof of Theorem 1. Furthermore, in the case , the index of is and one obtains at once that the number of subgroups having index () is , mentioned in Section 1.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Nicki Holighaus was partially supported by the Austrian Science Fund (FWF) START-Project FLAME (Y551-N13). László Tóth gratefully acknowledges support from the Austrian Science Fund (FWF) under Project no. M1376-N18. Christoph Wiesmeyr was partially supported by EU FET Open Grant UNLocX (255931).