Definition:Pullback
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Definition
Let $G, H$ be groups.
Let $N \triangleleft G, K \triangleleft H$.
Let $G / N \cong H / K$ such that $\theta: G / N \to H / K$ is such an 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 $\left({g, h}\right)$ where $\theta \left({g N}\right) = h K$.
This seems to me like a very superfluous way of stating 'preimage'. What is the added value of this concept?
Ans: Haven't a clue. Got the concept from the source below but at that stage I was getting out of my depth. Leave it in, though, it probably has subtleties.
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Also see
Sources
- John F. Humphreys: A Course in Group Theory (1996): $\S 13$: Definition $13.11$
