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

## Examples of Operations on Group Product

Solve for $x$ in:

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

## Solution

\(\displaystyle a x a^{-1}\) | \(=\) | \(\displaystyle a\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^{-1} a x a^{-1}\) | \(=\) | \(\displaystyle a^{-1} a\) | Group Product of both sides with $a^{-1}$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^{-1} a x a^{-1} a\) | \(=\) | \(\displaystyle a^{-1} a a\) | Group Product of both sides with $a$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x\) | \(=\) | \(\displaystyle a^{-1} a a\) | Group Axiom $G \, 3$: Inverses | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x\) | \(=\) | \(\displaystyle a\) | Group Axiom $G \, 3$: Inverses |

$\blacksquare$

## Sources

- 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Exercise $2 \ \text{(b)}$