Group Inverse-Seems to easy, please check

[Juanriq]

Group Inverse--Seems to easy, please check!

Hi all, this just seems too easy, and my teacher is prone to typos... I was thinking perhaps it was one.

Homework Statement

Prove that for a group G there exists a unique element that satisfies the equation x * x = x.


2. The attempt at a solution

Multiplying both sides by the inverse, x^{-1}, we have


x^{-1} * x * x = x^{-1} * x => (x^{-1} * x) * x = x^{-1} * x => e * x = e => x = e

and we see that for this relation to hold, x must be the identity element, which is unique.


Thanks in advance!

 

[JonF]

If you have already proved in class that the identity element must be unique it looks good to me.

 

[Juanriq]

Thanks!