We show that if G is a group and G has a graph-product decomposition with
finitely-generated abelian vertex groups, then G has two canonical
decompositions as a graph product of groups: a unique decomposition in which
each vertex group is a directly-indecomposable cyclic group, and a unique
decomposition in which each vertex group is a finitely-generated abelian group
and the graph satisfies the T0β property. Our results build on results by
Droms, Laurence and Radcliffe.Comment: 11 pages, 1 figur