Left and Right Inverses in an Abelian Group are the Unique Inverse

Last updated Nov 1, 2022

Suppose $G$ is an Abelian Group and $u, v \in G$. Further suppose $v$ is a Left Inverse/Right Inverse of $u$. Then $v$ is the unique inverse of $u$.

Because $G$ is an Abelian Group, we have by Commutativity that $vu = uv$. One of these will be $e$ by assumption, thus the other must be as well. Therefore $v$ is an inverse of $u$. Because The inverse in a Group is unique, we have uniqueness. $\blacksquare$