# Inverse Element is Power of Order Less 1

## Theorem

Let $G$ be a group whose identity is $e$.

Let $g \in G$ be of finite order.

Then:

$\order g = n \implies g^{n - 1} = g^{-1}$

## Proof

 $\displaystyle \order g$ $=$ $\displaystyle n$ $\displaystyle \leadsto \ \$ $\displaystyle g^n$ $=$ $\displaystyle e$ $\displaystyle \leadsto \ \$ $\displaystyle g^n g^{-1}$ $=$ $\displaystyle e g^{-1}$ $\displaystyle \leadsto \ \$ $\displaystyle g^{n - 1}$ $=$ $\displaystyle g^{-1}$

$\blacksquare$