Group Product Identity therefore Inverses
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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