On the homology of free abelianized extensions of finite groups

Abstract

Denote by F a free group of finite rank d with normal subgroup R of finite index and let G = F R . By a result of Kuz'min the homology groups H n( F R′ , Z) decompose into the direct sum of a free abelian group of finite rank d n and a finite abelian group. In this paper the ranks d n are computed and certain relations are derived. Moreover, it is shown that the Poincaré duality group F R′ is non-orientable if and only if d is even and the 2-Sylow subgroups of G are non-trivial and cyclic.