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The inverse in a Group is unique

Last updated Nov 1, 2022

# Statement

Suppose $G$ is a Group and $u, v, w \in G$ so that $v$, $w$ are both Additive Inverses of $u$. Then $v = w$.

# Proof

Denote $e \in G$ the Additive Identity. Then $$w = we = w(uv) = (wu)v = ev = v$$ by Associativity. $\blacksquare$