Pullbacks of Groups

Definition

Given three groups G₁,G₂,K and two epimorphims ψ_i : G_i ↠ K one can consider the pullback Γ of the G_i’s with kernel K given by the commutative diagram

define by Γ = {(g₁,g₂) ∈ G₁ × G₂ : ψ₁(g₁) = ψ(g₂)}. Observe that Γ is a subgroup or G₁ ×G₂.

Γ also can be viewed as an extension of K by the product of the kernels as follows

therefore the size of the pullback is |Γ| = |Ker(ψ₁)||Ker(ψ₂)||K|. There is a beautiful diagram maybe useful for cohomology