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. $\mathrm{\square }$

Title	uniqueness of inverse (for groups)
Canonical name	UniquenessOfInverseforGroups
Date of creation	2013-03-22 14:14:33
Last modified on	2013-03-22 14:14:33
Owner	waj (4416)
Last modified by	waj (4416)
Numerical id	5
Author	waj (4416)
Entry type	Result
Classification	msc 20-00
Classification	msc 20A05
Related topic	UniquenessOfAdditiveIdentityInARing
Related topic	IdentityElementIsUnique