# Thread: equivalence relation in a group

1. ## equivalence relation in a group

Let G be a group. Prove that the relation R on G, defined by xRy if and only if there exists an a that belongs to G such that a^-1*x*a, is an equivalence relation.

2. Originally Posted by anitra
Let G be a group. Prove that the relation R on G, defined by xRy if and only if there exists an a that belongs to G such that a^-1*x*a, is an equivalence relation.
you mean $xRy \Leftrightarrow y=a^{-1}xa$?

Do you know what an equivalence relation is? Which of the three properties are you having trouble proving?

3. Yes. that's exactly what I mean.
I do know what an equivalence relation is, but I got stuck at proving the reflexivity.

$x=a^{-1}xa$
is this trivial? Or how would I prove that?
Once I understand how to prove that, I think I'd be able to prove symmetry and transitivity.

4. Originally Posted by anitra
Yes. that's exactly what I mean.
I do know what an equivalence relation is, but I got stuck at proving the reflexivity.

$x=a^{-1}xa$
is this trivial? Or how would I prove that?
Once I understand how to prove that, I think I'd be able to prove symmetry and transitivity.
Can you think of an element $a \in G$ for which $x=a^{-1}xa$? Remember, you just have to show that there exists such an $a$, not that any choice of $a$ works (which, generally, is false).

5. ah, you got me there, you see, I'm having massive difficulty with group theory..

6. Originally Posted by anitra
Yes. that's exactly what I mean.
I do know what an equivalence relation is, but I got stuck at proving the reflexivity.

$x=a^{-1}xa$
is this trivial? Or how would I prove that?
Once I understand how to prove that, I think I'd be able to prove symmetry and transitivity.
You don't prove it. It isn't always true. What you want to prove is that there exist x such that $x= a^{-1}xa$.

What if a= e?