Unique Inverses

Theorem

Each element in a Group $G$ has a unique inverse.

Proof

Take any $a \in G$ and suppose $α_{1}, α_{2} \in G$ are both inverses of $a$ . Then:

α_{1} ⋆ a = α_{2} ⋆ a

Due to Cancellation Theorem (right cancellation) then $α_{1} = α_{2}$ .

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