A counterexample to questions on the integrality property of virtual signature

Wall (1961), defined the virtual Euler Characteristic χ(Γ) of an arbitrary group Γ of finite homological type as χ(Γ)=χ(Γ′)/[Γ:Γ′]∈Q, where Γ′ is any torsion free subgroup of finite index in Γ. Analogous to virtual Euler Characteristic, we define the Virtual signature of an oriented virtual Poincare Duality group, a rational number. We show that two of Ken Brownʹs results on questions regarding the integrality property of virtual Euler Characteristics when formulated in the Virtual signature case is false.