Inverse of Group Inverse/Proof 2

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Theorem

Let $\struct {G, \circ}$ be a group.

Let $g \in G$, with inverse $g^{-1}$.


Then:

$\paren {g^{-1} }^{-1} = g$


Proof

Let $g \in G$.

Then:

\(\displaystyle g g^{-1}\) \(=\) \(\displaystyle e\) Definition of Inverse Element
\(\displaystyle \leadsto \ \ \) \(\displaystyle g\) \(=\) \(\displaystyle \paren {g^{-1} }^{-1}\) Group Product Identity therefore Inverses

$\blacksquare$


Sources