example of fibre product

Let $G$, $G^{\prime }$, and $H$ be groups, and suppose we have homomorphisms $f:G\to H$ and $f^{\prime }:G^{\prime }\to H$. Then we can construct the fibre product $G{\times }_H{G}^{\prime }$. It is the following group:

$$\left\{(g,g^{\prime })\in G\times G^{\prime }\text{ such that }f(g)=f^{\prime }(g^{\prime })\right\}.$$

Observe that since $f$ and $f^{\prime }$ are homomorphisms, it is closed under the group operations.

Note also that the fibre product depends on the maps $f$ and $F^{\prime }$, although the notation does not reflect this.