On the homology of free Abelianized extensions of finite groups (Q1173680)

Let \(R \rightarrowtail F \twoheadrightarrow G\) be a finitely generated free presentation of the finite group \(G\). The group \(\Gamma := F/R'\) is an extension of \(G\) by the free abelian group \(R/R'\), and hence a Poincaré duality group of dimension \(m\) where \(m = \text{rank }R/R'\). Using the Lyndon-Hochschild-Serre spectral sequence the author deduces a formula giving the rank \(h_ n\) of the homology group \(H_ n(\Gamma,\mathbb{Z})\). Somewhat surprisingly, the only structural properties of \(G\) involved in this formula are the numbers of elements in \(G\) of order precisely \(q\) for the divisors \(q\) of \(n\) and \(n-1\). Finally, the author shows that \(\Gamma\) is non orientable iff the number of generators of \(F\) is even and the 2-Sylow subgroups of \(G\) are non trivial and cyclic.