inverse of inverse in a group

Let (G,*) be a group. We aim to prove that (a-1)-1=a for every aG. That is, the inverse of the inverse of a group element is the element itself.

By definition a*a-1=a-1*a=e, where e is the identity in G. Reinterpreting this equation we can read it as saying that a is the inverse of a-1.

In fact, consider b=a-1, the equation can be written a*b=b*a=e and thus a is the inverse of b=a-1.

Title inverse of inverse in a group
Canonical name InverseOfInverseInAGroup
Date of creation 2013-03-22 15:43:36
Last modified on 2013-03-22 15:43:36
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 7
Author cvalente (11260)
Entry type Proof
Classification msc 20-00
Classification msc 20A05
Classification msc 08A99
Related topic AdditiveInverseOfAnInverseElement