\begin{definition}[Definition:Pullback of Quotient Group Isomorphism]
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 $\map \theta {g N} = h K$.
\end{definition}