inverse of a product


If a and b are arbitrary elements of the group  (G,*), then the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of a*b is

(a*b)-1=b-1*a-1. (1)

Proof.  Let the neutral elementPlanetmathPlanetmath of the group, which may be proved unique, be  e.  Using only the group postulatesMathworldPlanetmath we obtain



Note.  The (1) may be by inductionMathworldPlanetmath extended to the form

Title inverse of a productPlanetmathPlanetmath
Canonical name InverseOfAProduct
Date of creation 2015-01-30 21:19:19
Last modified on 2015-01-30 21:19:19
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Theorem
Classification msc 20A05
Classification msc 20-00
Synonym inverse of a product in group
Synonym inverse of product
Related topic InverseOfCompositionOfFunctions
Related topic GeneralAssociativity
Related topic Division
Related topic InverseNumber
Related topic OrderOfProducts