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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. $\blacksquare$

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Group

Group

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Left and Right Inverses in an Abelian Group are the Unique Inverse

Left and Right Inverses in an Abelian Group are the Unique Inverse

Statement Suppose $G$ is an and $u, v \in G$ . Further suppose $v$ is a / of $u$ . Then $v$ is the unique inverse of $u$ . Proof Because $G$ is an , we have by that $vu =......

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