---- > [!definition] Definition. ([[cokernel of an abelian group homomorphism]]) > [[categorical cokernel|Cokernels]] exist in the [[category]] $\mathsf{Ab}$ of [[abelian group|abelian groups]]. > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAHEQBfU9TXfIRQBGclVqMWbdgHJuvEBmx4CRAExjq9Zq0QgAMvL7LBRUcPHapegDo2cMAB45gAYwgBrGACcABF187em80AAssbnEYKABzeCJQADNvCABbJDIQHAgkUQkdNiC6EPCQagY6ACMYBgAFfhUhEG8sGNCcIxBktNzqbKQNfOsQO0YwujKQCuq6htM9FraOniSU9MRM-sRB6rAoJABmTKtdEGJO7vW8rYPqUJg6fb0cAHcIe8eELUlTuzQIlZdNaHPo5bblLBgU5QOhwe77b4FWw2JxYOA4OC+ACEgRsEBoPgYkJgwFGDHGXEm0xq9RMqgWrXakS4QA > \begin{tikzcd} > G \arrow[r, "\varphi"'] \arrow[rr, "0", bend left] & G' \arrow[r, "\alpha"'] \arrow[d, "\pi", two heads] & L \\ > & \text{coker } \varphi \arrow[ru, "\exists ! \overline{\alpha}"', dashed] & > \end{tikzcd} > \end{document} > ``` > Explicitly, if $G,G'$ are [[abelian group|abelian groups]] and $\varphi:G \to G'$ a [[group homomorphism]], then $\text{coker }\varphi \cong \frac{G'}{\im \varphi}$. ^definition > [!justification] > We need to show that the defining [[universal property]] of [[categorical cokernel|cokernels]] is satisfied. Since $G'$ is [[abelian group|abelian]], $\im \varphi$ is a [[normal subgroup]] of $G'$. Then the condition that $\alpha \circ \varphi$ is trivial says that $\im \varphi \subset \ker \alpha$, and hence $\frac{G'}{\im \varphi}$ satisfies the [[universal property]] (using the [[characterization of quotienting a group|universal property of quotient groups]]). ^justification ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY 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 > ```