Let Fn be the free group of rank n with basis x1, x2, …, xn, and let d(G) denote the minimal number of generators of the finitely generated group G. Suppose that n[ges ]d(G). There exists an exact sequence

formula here

and we may view the free abelian group R=R/R′ as a right ℤG-module by defining (rR′)g=rgϕ−1R′ for all g∈G, where gϕ−1 is any preimage of g under ϕ, and rgϕ−1=(gϕ−1)−1r(gϕ−1), the conjugate of r by gϕ−1. We call R the relation module of G associated with the presentation (1), and say that R has ambient rank n. Furthermore, we call the group Fn/R′ the free abelianized extension of G associated with (1).