# Product with Inverse equals Identity iff Equality

## Theorem

Let $\struct {G, \circ}$ be a group whose identity element is $e$.

Then:

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

## Proof

Using various properties of groups:

 $\displaystyle a \circ b^{-1}$ $=$ $\displaystyle e$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \paren {a \circ b^{-1} } \circ b$ $=$ $\displaystyle e \circ b$ Cancellation Laws $\displaystyle \leadstoandfrom \ \$ $\displaystyle a \circ \paren {b^{-1} \circ b}$ $=$ $\displaystyle e \circ b$ Group Axiom $\text G 1$: Associativity $\displaystyle \leadstoandfrom \ \$ $\displaystyle a$ $=$ $\displaystyle b$ Group Axiom $\text G 2$: Existence of Identity Element and Group Axiom $\text G 3$: Existence of Inverse Element

$\blacksquare$