# Doubt about abelian conjugacy class

1. Mar 24, 2015

### frank1

1. The problem statement, all variables and given/known data
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 = hg

Then it says that if the group is abelian h = hg

2. Relevant equations
Abelian group: a*b = b*a

3. 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 = hg

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 aswell then leading to every non-abelian group also having the property h = hg?

PS: Sorry my english.

2. Mar 24, 2015

### 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.

3. Mar 24, 2015

### frank1

got it! thanks fourier