uniqueness of inverse (for groups)

Lemma Suppose (G,) is a group. Then every element in G has a unique inverseMathworldPlanetmathPlanetmathPlanetmath.

Proof. Suppose gG. By the group axioms we know that there is an hG such that


where e is the identity elementMathworldPlanetmath in G. If there is also a hG satisfying




so h=h, and g has a unique inverse.

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