Doubt about abelian conjugacy class

[frank1]

Homework Statement

I'm kinda lost in the concept of conjugate elements in group theory. It says that a element "h conjugate by g" is:

g^-1×h×g = h^g

Then it says that if the group is abelian h = h^g

Homework Equations

Abelian group: a*b = b*a

The Attempt at a Solution

I don't get why the fact that the group is abelian (a*b = b*a) leads to the conclusion that h = h^g

g^-1×h×g = g×h×g^-1 why does it lead to h. I know it implies g×g^-1 = e. But why wouldn't g^-1×g = e as well then leading to every non-abelian group also having the property h = h^g?

PS: Sorry my english.

 

[fourier jr]

if the group is abelian then it's possible to swap the $g^{-1}$ & $h$ or the $h$ & $g$ to get either $g^{-1}g$ or $gg^{-1}$ leaving h by itself.

 

[frank1]

got it! thanks fourier