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 $g = h^{-1}$


Sources

  • 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 35.3$: Elementary consequences of the group axioms
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