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

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 $\quad$

$\blacksquare$