---- > [!definition] Definition. ([[group homomorphism]]) > A **homomorphism of [[group]]s** $\varphi:G \to H$ is a map satisfying $\varphi(ab)=\varphi(a)\varphi(b)$ for all $a,b \in G$, i.e. a map $\varphi$ for which the following diagram commutes: > \ > The **trivial homomorphism** $\varphi_{\text{trivial}}$ is the map which sends every $a \in G_{1}$ to $e_{G_{2}}$. It makes $\text{Hom}_{\mathsf{Grp}}(G_{1},G_{2})$ a [[pointed set]]. > > > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAHEACAHW7wFt4nLiAC+pdJlz5CKAIzkqtRizYAJHnyyC4nNWIkgM2PASJk5S+s1aIOBySZlEFl6tdV39opTCgBzeCJQADMAJwh+JDIQHAgkBWUbNl56MLQACyxNASFUunSshxBwyOjqOKQAJncVWxBeAGMoCBwAfXYQagY6ACMYBgAFKVNZEDCsfwycYtKoxETKxABmWuS7Jpb270M56or4lbXPBu40zKwukB7+oZHnOwmpmZ9RIA > \begin{tikzcd} > G \times G \arrow[r, "\varphi \times \varphi"] \arrow[d, "\cdot_G"'] & H \times H \arrow[d, "\cdot_H"] \\ > G \arrow[r, "\varphi"'] & H > \end{tikzcd} > \end{document} > ``` > > [!basicexample] > - [[det is a group homomorphism]] > - The [[modulus|modulus map]] $|\cdot|:\mathbb{C}^{\times} \to \mathbb{R}^{\times}$ is a [[group homomorphism]]. See [[complex numbers#Properties of **complex numbers**]]. > - The exponential map $\text{exp}:\mathbb{R}^{+} \to \mathbb{R}^{\times}$ is a [[group homomorphism]] because $e^{a+b}=e^{a}e^{b}$. > [!basicproperties] > - [[group homomorphisms take identities to identities and inverses to inverses]] > - [[group homomorphism is injective iff kernel is trivial iff is a monomorphism]] > **More generally:** [[group homomorphisms preserve structure]] ---- #### ---- #### 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 > ```