---- > [!theorem]+ Theorem. ([[universal property of group homomorphism kernels]]) > ~ > Let $G,G'$ be [[group|groups]] and $\varphi: G \to G'$ a [[group homomorphism]]. > > Let $\mathsf{C}$ be the [[subcategory]] of the [[slice category]] $\mathsf{Grp}_{G}$ obtained by keeping as objects only the [[group homomorphism|group homomorphisms]] $\alpha:K \to G$ satisfying $\varphi \circ \alpha=\text{trivial map}$. > > Then the [[inclusion map]] $\iota: \ker \varphi \hookrightarrow G$ is [[terminal object|final]] in $\mathsf{C}$. In other words, every [[group homomorphism]] $\alpha:K \to G$ such that $\varphi \circ \alpha$ is [[group homomorphism|trivial]] factors uniquely through $\text{ker } \varphi$: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGkQBfU9TXfIRQBGclVqMWbAOLdeIDNjwEiAJjHV6zVohDSA5HL5LBRUcPFapugDo2cMAB45gAaxgAnAARc79D2gAFljc4jBQAObwRKAAZh4QALZIZCA4EEgAzJqSOiB2TlhwOHBeAIRedhA0ngxYYDDAdoxBdFwg1Ax0AEYwDAAK-MpCIB5YEYE4RiDxSUiiaRmI6hLabH50AcHTs8mIqenzOWu2Ni2BdDsJewdLK71gUFmpVnl2Ds7AOGM0WIztPDi1yy1EOiAWr3WNnwOEuXAoXCAA > \begin{tikzcd} > K \arrow[rd, "\exists ! \overline{\alpha}"'] \arrow[r, "\alpha"] \arrow[rr, "\text{trivial}", bend left] & G \arrow[r, "\varphi"] & G' \\ > & \text{ker }\varphi \arrow[u, "\iota"] & > \end{tikzcd} > \end{document} > ``` ^theorem > [!generalization] > The note [[categorical kernel]] defines, for a certain type of [[category]], the notion of a 'kernel' via a [[universal property]]. ^generalization > [!proof]+ Proof. ([[universal property of group homomorphism kernels]]) > If $\alpha: K \to G$ is such that $\varphi \circ \alpha$ is [[group homomorphism|trivial]], then for all $k \in K$, $\varphi \circ \alpha(k)=\varphi(\alpha(k))=e_{G'},$ > that is, $\alpha(k) \in \text{ker }\varphi$. So we can (and must) let $\overline{\alpha}$ be $\alpha$ itself. ^proof ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch01