# inverse of inverse in a group

Let $(G,*)$ be a group. We aim to prove that ${(a^{-1})}^{-1}=a$ for every $a\in G$. 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 InverseOfInverseInAGroup 2013-03-22 15:43:36 2013-03-22 15:43:36 cvalente (11260) cvalente (11260) 7 cvalente (11260) Proof msc 20-00 msc 20A05 msc 08A99 AdditiveInverseOfAnInverseElement