----- - [ ] these notes are all a bit too atomic, this should go in [[group homomorphisms preserve structure]] > [!proposition] Proposition. ([[group homomorphisms take identities to identities and inverses to inverses]]) > Let $\phi:G \to H$ be a [[group homomorphism]]. Then $\phi(e_{G})=e_{H}$ and $\phi(g ^{-1})=\phi(g)^{-1}$. > [!proof]- Proof. ([[group homomorphisms take identities to identities and inverses to inverses]]) > First we have $\phi(e_{G})=\phi(e_{G} e_{G})=\phi(e_{G})\phi(e_{G})=(\phi(e_{G}))^{2}.$ > Second we have $\phi(g ^{-1})\phi(g)=\phi(g^{-1} g)=e_{H}$ > which happens if and only if $\phi(g ^{-1})=\phi(g)^{-1}.$ > ----- #### ---- #### 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 > ```