# Inverse of Group Inverse

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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 1

Let $g \in G$.

Then:

\(\displaystyle g\) | \(\in\) | \(\displaystyle G\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle e\) | \(=\) | \(\displaystyle g^{-1} \circ g\) | Definition of Inverse Element | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {g^{-1} }^{-1} \circ e\) | \(=\) | \(\displaystyle \paren {g^{-1} }^{-1} \circ \paren {g^{-1} \circ g}\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {g^{-1} }^{-1} \circ e\) | \(=\) | \(\displaystyle \paren {\paren {g^{-1} }^{-1} \circ g^{-1} } \circ g\) | Definition of Associative Operation | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {g^{-1} }^{-1} \circ e\) | \(=\) | \(\displaystyle e \circ g\) | Definition of Inverse Element | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {g^{-1} }^{-1}\) | \(=\) | \(\displaystyle g\) | Definition of Identity Element |

$\blacksquare$

## Proof 2

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

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{II}$: The Group Property

- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups: Theorem $1.2 \text{(iv)}$