---- > [!definition] Definition. ([[group embedding]]) > An **embedding** of a [[group]] $G_{1}$ into a [[group]] $G_{2}$ is an [[injection|injective]] [[group homomorphism]] $G_{1}\to G_{2}$, i.e., a [[group homomorphism]] that is an [[group isomorphism|isomorphism]] onto its image. > [!note] Remark. > In general [[topological space|topological spaces]], not all [[continuous]] [[bijection|bijections]] have [[continuous]] [[inverse map|inverses]], and hence not all [[continuous]] [[bijection|bijections]] are [[isomorphism|isomorphisms]] onto their images. In contrast, the inverse of a [[group isomorphism]] is guaranteed to also be a [[group isomorphism]]: see [[group isomorphism#^556381|here]]. This is why in [[topological embedding]] we are not able to merely say 'an injective [[continuous]] map' (like we do here replacing [[continuous]] map with [[group homomorphism|homomorphism]]). ^note ---- #### ---- #### 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 > ```