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

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Examples of Operations on Group Product

Solve for $x$ in:

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


Solution

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

$\blacksquare$


Sources