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Let F n denote the free group of rank n and d(G) the minimal number of generators of the finitely generated group G. Suppose that R ↪ F m ↠ G and S ↪ F m ↠ G are presentations of G and let $$\overline{R}$$ and $$\overline{S}$$ denote the associated relation modules of G. It is well known that $$\overline{R}\oplus {\left(\mathbb{Z}G\right)}^{d\left(G\right)}\cong \overline{S}\oplus {\left(\mathbb{Z}G\right)}^{d\left(G\right)}$$ even though it is quite possible that . However, to the best of the author’s knowledge no examples have appeared in the literature with the property that . Our purpose here is to exhibit, for each integer k ≥ 1, a group G that has presentations...

A subgroup H of a group G is inert if |H: H ∩ H g| is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally graded simple groups cannot be totally inert.