Product with Inverse equals Identity iff Equality

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Theorem

Let $\left({G, \circ}\right)$ be a group whose identity element is $e$.

Then:

$\forall a, b \in G: a \circ b^{-1} = e \iff a = b$

Using various properties of groups:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle a \circ b^{-1}$	$=$	$\displaystyle e$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \left({a \circ b^{-1} }\right) \circ b$	$=$	$\displaystyle e \circ b$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Cancellation Laws
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle a \circ \left({b^{-1} \circ b}\right)$	$=$	$\displaystyle e \circ b$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Associativity of $\circ$ in a group
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle a$	$=$	$\displaystyle b$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Properties of identity and inverse

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a \circ b^{-1}\)	\(=\)	\(\displaystyle e\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \left({a \circ b^{-1} }\right) \circ b\)	\(=\)	\(\displaystyle e \circ b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Cancellation Laws
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle a \circ \left({b^{-1} \circ b}\right)\)	\(=\)	\(\displaystyle e \circ b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity of $\circ$ in a group
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle a\)	\(=\)	\(\displaystyle b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Properties of identity and inverse