# Group Product/Examples/a x a^-1 = a

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## Examples of Operations on Product Elements

Solve for $x$ in:

$a x a^{-1} = a$

## Solution

 $\displaystyle a x a^{-1}$ $=$ $\displaystyle a$ $\displaystyle \leadsto \ \$ $\displaystyle a^{-1} a x a^{-1}$ $=$ $\displaystyle a^{-1} a$ Product of both sides with $a^{-1}$ $\displaystyle \leadsto \ \$ $\displaystyle a^{-1} a x a^{-1} a$ $=$ $\displaystyle a^{-1} a a$ Product of both sides with $a$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle a^{-1} a a$ Group Axiom $\text G 3$: Existence of Inverse Element $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle a$ Group Axiom $\text G 3$: Existence of Inverse Element

$\blacksquare$