Decomposition of Goursat Matrices and Subgroups of Z m × Z n

Given the number of subgroups of Z m × Z n , we deduce the Goursat matrix. The purpose of this paper is two-fold. A ﬁrst and more concrete aim is to demonstrate that the triangular decomposition of the Goursat matrix may also be written out explicitly, and furthermore that the same is true of the inverse of these triangular factors. A second and more abstract aim provides a containment relation property between subgroups of a direct product . Namely, if U 2 ≤ U 1 ≤ Z m × Z n , we provide necessary and sufﬁcient conditions for U 2 ≤ U 1 .


Introduction
One of the most major problems of the combinatorial abelian group theory is to investigate the number of subgroups of a finite abelian group. This topic has enjoyed a constant evolution starting with the first half of the 20 th century. In 1897 Goursat proved that every subgroup of the direct product of two groups is determined by an isomorphism between factors groups of the given groups [8]. Given Goursat's lemma for groups, we use as consequences, by purely number theoretical arguments, explicit formulas for the total number of subgroups of Z m × Z n and f p (i, j) the number of subgroups of Z p i ×Z p j . This allows us to deduce a new type of symmetric matrix G p (n) with f p (i, j) i,j as coefficient, also called Goursat's matrix of order n + 1. Expressing a matrix as a product of a lower triangular matrix L and an upper triangular matrix U is called an LU factorization. Such a factorization is typically obtained by Gaussian elimination. If L is a lower triangular with unit main diagonal and U is an upper triangular, the LU Definition 2.2. For a subgroup U of L × R. We say that the corresponding 5-tuple Q 5 (U ) = (A, B, C, D, θ) of Lemma 2.1 is the Goursat decomposition of U .

The LU Factorization of the Goursat Matrix
In the following let us denote by f p (i, j) the total number of all subgroups of the finite abelian p-group Z p i × Z p j , (i ≤ j), concerning general properties of the subgroup lattice of finite Abelian groups, see Tòth [22], Tãrnãuceanu [25]. For every p ∈ P one has Put f p (i, j) = f p (j, i), for all i > j, and let n be a fixed positive integer. We denote the Goursat matrix of order n + 1  by induction on n one easily obtains , for any n ≥ 1.
An L p (n)U p (n) decomposition of a matrix G p (n) is the product of lower triangular matrix and an upper triangular matrix that is equal to G p (n).
Theorem 3.2. Given the Goursat matrix of order n + 1, where G p (n) = L p (n)U p (n). Then the triangular factors are Proof. It is convenient to use the following notation for the general square matrix A of order k (see [6]) We denote the determinant of this matrix by |a 1 b 2 c 3 · · · w k | . Thus It is known (Turnbull [27], p.369) that for the usual triangular decomposition A = LU, the elements of L and U can be expressed in terms of determinants involving the elements of A as follows (it is assumed that the decomposition is possible and L has been chosen to have units in the principal diagonal) : It does not seem to have been noticed that L −1 p (n) and U −1 p (n) can similarly be expressed explicitly as follows : Theorem 3.4. Given the Goursat matrix of order n + 1, In some problems one wishes to decompose a matrix into a product of the form U L rather than in the more usual form LU .
In the following let us denote by h p (i, j) the number f p (i, j) − i − j. Let n be a fixed positive integer and H p (n) be the matrix (h p (i, j)) 0≤i,j≤n . Then H p (n) induces a quadratic form by induction on n one easily obtains , for any n ≥ 1.
Hence, we have proved the next corollary.
Corollary 3.9. Given the matrix of order n + 1, (h p (i, j)) 0≤i,j≤n where H p (n) = 4 Containment of Subgroups of a Direct Product Z m × Z n Throughout the paper we use the following notation: N := {1, 2, · · · }, P is the set of primes; the prime power factorization of n ∈ N is n = p∈P p νp(n) , where all but a finite number of the exponents ν p (n) are zero; Z n denotes the additive group of residue classes modulo n (n ∈ N); a ∧ b and a ∨ b denote the greatest common divisor and least common multiple, respectively of a, b. For every m, n ∈ N, let J m,n := (a, b, c, d, t) ∈ N 5 : a|m, b|a, c|n, d|c, Using the condition a b = c d .
Step 2. Writing explicitly the corresponding subgroups and quotient groups we and And similarly Corollary 4.2. Let m, n ∈ N and U 2 , U 1 ≤ Z m × Z n . Then Proof.
Definition 4.3. Let G be a direct product of Z m × Z n . Given a chain of subgroups of G of the form We say that k is the length of the chain and x k the chain. Let k = ε(m, n), and ω(m, n) := |{x k 1 , x k 2 , · · · }| the number of the chain.
Consider G = Z 5 × Z 25 . See Figure for the subgroup lattice of G.

Acknowledgments
The author wishes to sincerely thank the referees for several useful comments.