uniqueness of inverse (for groups)

LemmaSuppose $(G,\ast )$ is a group. Then every element in $G$ has aunique inverse.

Proof. Suppose $g\in G$. By the group axioms we know that thereis an $h\in G$ such that

$$g\ast h=h\ast g=e,$$

where $e$ is the identity element in $G$. If there is also a $h^{\prime }\in G$satisfying

$$g\ast h^{\prime }=h^{\prime }\ast g=e,$$

then

$$h=h\ast e=h\ast (g\ast h^{\prime })=(h\ast g)\ast h^{\prime }=e\ast h^{\prime }=h^{\prime },$$

so $h=h^{\prime }$, and $g$ has a unique inverse. $\mathrm{\square }$