Definition:Pullback of Quotient Group Isomorphism

Definition
Let $G, H$ be groups.

Let $N \lhd G, K \lhd H$ be normal subgroups of $G$ and $H$ respectively.

Let:
 * $G / N \cong H / K$

where:
 * $G / N$ denotes the quotient of $G$ by $N$
 * $\cong$ denotes group isomorphism.

Let $\theta: G / N \to H / K$ be such a group isomorphism.

The pullback $G \times^\theta H$ of $G$ and $H$ via $\theta$ is the subset of $G \times H$ of elements of the form $\tuple {g, h}$ where $\map \theta {g N} = h K$.

Also see

 * Pullback of Quotient Group Isomorphism is Subgroup