Note that previous assumption leads to the conclusion that chain map C k (K/G; F) → C k (X; F) Γ is an isomorphism. Additionally, results from above can be gereralized to relative homology groups, that H k (K, L; F) Γ ∼ = H k (K/Γ, L/Γ; F) where L ⊂ K is Γ-invariant. Proof for this widely known theorem can be found in some previous print collections .