2007/01/08 by K. W. Gruenberg, Karl W. Gruenberg, Gruenberg, Karl W. +2
Computer Science · Mathematics · #20F05 (Primary) 20E06 (Secondary) #Advanced Graph Theory Research #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20E06 #msc:20F05 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0701225
18 pages, minor changes. To appear in J. Group Theory
openalex publication_date 2007/01/08 · arxiv created 2007/12/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a group of the form G1* ... *Gn, the free product of n subgroups, and let M be a ZG-module of the form \bigoplusi=1n Mi ⊗ℤGi ℤG. We shall give formulae in various situations for dZG(M), the minimum number of elements required to generate M. In particular if C1,C2 are non-trivial finite cyclic groups of coprime orders, G = (C1 × Z) * (C2 × Z) and F/R ≅ G is the free presentation obtained from the natural free presentations of the two factors, then the number of generators of the relation module, dℤG(R/R') is three. It seems plausible that the minimum number of relators of G should be 4, and this would give a finitely presented group with positive relation gap. However we cannot prove this last statement.