Inverse of Group Product/Proof 2
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Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $a, b \in G$, with inverses $a^{-1}, b^{-1}$.
Then:
- $\paren {a \circ b}^{-1} = b^{-1} \circ a^{-1}$
Proof
We have that a group is a monoid, all of whose elements are invertible.
The result follows from Inverse of Product in Monoid.
$\blacksquare$