Relation modules of abelian groups

Abstract

be a free presentation for G, where F is a finitely generated free group. Then R = R/R' is a 7/G-module, called the relation module associated with the presentation (1). We shall say that/~ is a minimal relation module if d (F) = d (G), where d (G) denotes the minimum number of elements required to generate G. For any positve integer d, define r d (G) to be the number of isomorphism classes of relation modules for G with d (F) = d. We shall abbreviate rd(G)(G) to r(G); thus r(G) is the number of minimal relation modules for G. In this paper we are concerned with calculating r d (G) for when G is abelian. To state our results, we need to establish some notation. For any finitely generated abelian group H, let h (H) denote its Hirsch number, that is its torsion free rank. If a, b ~ ~, then a [b means that a divides b. We shall also use this notation when a or b is oo; our convention will be that al oo for all a, and oo Ib implies b = oo. We shall write 2 (G) for the highest common factor of the torsion coefficients of the finitely generated abelian group G; thus if G = C 1 x C 2 x ..- x C d with the C z cyclic of order nl (possibly oo) and 1 d (G), then r d (G) = 1. (ii) If h (G) > 2, then r (G) = v (2 (G)). (iii) Ifh(G) = 1 or 2, then r(G) = ~(d(G) - 1, 2(G)).