----- > [!proposition] Proposition. ([[group of order 1-2 elements is abelian]]) > If $G$ is a [[group]] whose elements all square to $e$, then $G$ is [[abelian group|abelian]]. > [!proof]- Proof. ([[group of order 1-2 elements is abelian]]) > If $G$ has [[order of a group|order]] $1$ or $2$ it's obvious, so suppose $|G| \geq 3$. Take nontrivial $a,b \in G$. We have $a^{2}b^{2}=\textcolor{Skyblue}{a(a b)b}=e=(ab)(ab)=\textcolor{Skyblue}{a(ba)b},$ > now apply [[cancellation law for groups|cancellation]] to the blue terms. ^77cb24 ----- #### ---- #### 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 > ```