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

$\blacksquare$


Sources