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

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Examples of Operations on Group Product

Solve for $x$ in:

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


Solution

\(\displaystyle a x a^{-1}\) \(=\) \(\displaystyle e\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle a^{-1} a x a^{-1}\) \(=\) \(\displaystyle a^{-1} e\) Group Product of both sides with $a^{-1}$
\(\displaystyle \leadsto \ \ \) \(\displaystyle a^{-1} a x a^{-1} a\) \(=\) \(\displaystyle a^{-1} e a\) Group Product of both sides with $a$
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle a^{-1} e a\) Group Axiom $G \, 3$: Inverses
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle a^{-1} a\) Group Axiom $G \, 2$: Identity
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle e\) Group Axiom $G \, 3$: Inverses

$\blacksquare$


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