# 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.

Title example of fibre product ExampleOfFibreProduct 2013-03-22 14:08:38 2013-03-22 14:08:38 archibal (4430) archibal (4430) 4 archibal (4430) Example msc 14A15 Group Homomorphism CartesianProduct