20-00-UniquenessOfInverseforGroups.tex

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\pmtitle{uniqueness of inverse (for groups)}
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\pmrelated{UniquenessOfAdditiveIdentityInARing}
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\begin{document}
\PMlinkescapeword{lemma}

{\bf Lemma}
Suppose $(G,\ast)$ is a group. Then every element in $G$ has a
unique inverse.

\emph{Proof.} Suppose $g\in G$. By the group axioms we know that there
is an $h\in G$ such that 
$$ g\ast h = h\ast g = e,$$
where $e$ is the identity element in $G$. If there is also a $h'\in G$
satisfying
$$ g\ast h' = h'\ast g = e,$$
then 
$$ h = h\ast e = h \ast(g \ast h') =(h\ast g)\ast h' = e\ast h' = h',$$
so $h=h'$, and $g$ has a unique inverse. $\Box$
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