----- > [!proposition] Proposition. ([[squaring map is a homomorphism iff group is abelian]]) > The map $g \mapsto g ^2$ is a [[group homomorphism]] iff $G$ is an [[abelian group]]. ^proposition > [!proof]- Proof. ([[squaring map is a homomorphism iff group is abelian]]) > ~ > Let $g \xmapsto{\phi}g^{2}$. Suppose $G$ is [[abelian group|abelian]], then $\phi(gh)=(gh)^{2}=ghgh=g^{2}h^{2}=\phi(g)\phi(h)$ so $\phi$ is a [[group homomorphism]]. > Suppose $G$ is not an [[abelian group]]; fix $g,h$ for which $gh \neq hg$. If $\phi(gh)=(gh)^{2}=ghgh=\phi(g)\phi(h)=gghh$, then [[cancellation law for groups|cancellation]] implies $hg=gh$. So $\phi$ must not be a [[group homomorphism]]. ^308f3f ----- #### ---- #### 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 > ```