Relation modules of Abelian groups
Let G be a finitely generated group and let 1\(\to R\to F\to G\to 1\) be a free presentation for G, where F is a free group of finite rank k. Then the \({\mathbb{Z}}G\)-lattice \(\bar R=R/R'\) is called the relation module associated to this free presentation. Define \(r_ k(G)\) to be the number of isomorphism classes of relation modules for G with \(rank(F)=k\). The main result of the article is a complete description of \(r_ k(G)\) for any finitely generated infinite abelian group G. More precisely, let d be the minimum number of elements of the finitely generated abelian group G required to generate G. Denote by h the Hirsch number of G (i.e. its torsion free rank) and let t be the greatest positive integer such that the d-fold cartesian power of the additive group of the ring \({\mathbb{Z}}_ t={\mathbb{Z}}/t{\mathbb{Z}}\) is isomorphic to a subgroup of G. Finally let \({\mathbb{Z}}^*_ t\) be the unit group of \({\mathbb{Z}}_ t ({\mathbb{Z}}^*_ t=\{1\}\) if \(t=1)\). Then (i) If \(k>d\), then \(r_ k(G)=1\). (ii) If \(h>2\), then \(r_ d(G)=| {\mathbb{Z}}_ t^*/<\pm 1>|\). (iii) If \(h=1\) or 2, then \(r_ d(G)=| {\mathbb{Z}}^*_ t/\pm ({\mathbb{Z}}_ t^*)^{d- 1}|\). It should be remarked that not for every finite abelian group such a precise formula is known [cf. \textit{P. J. Webb}, J. Pure Appl. Algebra 21, 205-232 (1981; Zbl 0488.20007)].
finitely generated infinite abelian group
scientific article; zbMATH DE number 3869567