inverse of a product


Theorem.

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

(a*b)*(b-1*a-1)=a*(b*(b-1*a-1))=a*((b*b-1)*a-1)=a*(e*a-1)=a*a-1=e,
(b-1*a-1)*(a*b)=b-1*(a-1*(a*b))=b-1*((a-1*a)*b)=b-1*(e*b)=b-1*b=e,

Q.E.D.

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

(a1**an)-1=an-1**a1-1.
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