Proof about identity element of a group

[hoopsmax25]

Homework Statement

If G is a group, a is in G, and a*b=b for some b in G (* is a certain operation), prove that a is the identity element of G

Homework Equations

The Attempt at a Solution

I feel like you should assume a is not the identity element and eventually show that a= the identity. but I am not sure how to show that.

 

[LCKurtz]

Hint: Think about using $b^{-1}$.

 

[hoopsmax25]

I thought about using the inverse of b but I'm not sure how to plug it in?

 

[LCKurtz]

You don't "plug it in". You could try *-ing things with it.

 

[hoopsmax25]

Could you do a*b*b-inverse=b*b-inverse? and go from there?