# Group Product Identity therefore Inverses

## Theorem

Let $g$ and $h$ be elements of a group $G$ whose identity element is $e$.

Then if either:

$g h = e$

or:

$h g = e$

it follows that:

$g = h^{-1}$

and:

$h = g^{-1}$

### Part 1

$g h = e \implies h = g^{-1}$ and $g = h^{-1}$

### Part 2

$h g = e \implies h = g^{-1}$ and $g = h^{-1}$