Subgroups of Finite Abelian Groups Having Rank Two via Goursat’s Lemma

Abstract

Using Goursat's lemma for groups, a simple representation and the invariant factor decompositions of the subgroups of the group Z m × Z n are deduced, where m and n are arbitrary positive integers. As consequences, explicit formulas for the total number of subgroups, the number of subgroups with a given invariant factor decomposition, and the number of subgroups of a given order are obtained.

A generalized Goursat lemma

“…Other sources that mention Goursat's lemma include papers of S. Dickson [Dic69] in 1969, K. Ribet [Rib76] in 1976, a book by R. Schmidt [Sch94, Chapter 1.6], a paper of D. Anderson and V. Camillo in 2009 [AC09], a preprint of A. Greicius in 2009 [Gre09], a recent preprint by L. Tóth [Tóth14], and several internet sites such as [FL10,AEM09]. Taken together, these various sources demonstrate the applicability of Goursat's lemma to diverse branches of mathematics.…”

Decomposition of Goursat Matrices and Subgroups of Zm x Zn

“…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…”

On the number of subgroups of a given exponent in a finite abelian group

“…The first one is based on the known formula for the number of subgroups of a given type in an abelian p-group, given in terms of gaussian coefficients (Theorem 2.1). The second method, applicable only for rank two groups, uses the representation of the subgroups of Z m × Z n obtained by the second author in [14] (Theorem 3.1).…”

“…The total number s(m, n) of subgroups of the group Z m × Z n is (see[6, Th. 3],[14, Th. 4.1]) ) shows the distribution of the number of subgroups according to their exponents.…”