# uniqueness of inverse (for groups)

Lemma Suppose $(G,\ast)$ is a group. Then every element in $G$ has a unique inverse.

Proof. Suppose $g\in G$. By the group axioms we know that there is 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. $\Box$

Title uniqueness of inverse (for groups) UniquenessOfInverseforGroups 2013-03-22 14:14:33 2013-03-22 14:14:33 waj (4416) waj (4416) 5 waj (4416) Result msc 20-00 msc 20A05 UniquenessOfAdditiveIdentityInARing IdentityElementIsUnique