# Isotropy Group and equivalent relation

• Oct 24th 2010, 05:44 AM
Sogan
Isotropy Group and equivalent relation
Hello,

my english is not so good, but i try my best to explain my problems to you.

I have a problem with following situation:

Let G be a group, A a arbitrary set, $\displaystyle f:G x A->A$ a group action and $\displaystyle G_a$ the isotropy group of a $\displaystyle \in A$.

Now my professor has denoted followin equiv. relation on G:

g ~ g'<=>ga=g'a<=>g'^(-1)g=g^(-1)g' $\displaystyle \in G_a <=> g=g'mod G_a$=>Ga~G/G_a

And my question is, what does Ga~G/G_a means? i never had seen this before.

I hope you can understand my problem. If not, please tell me which part you dont understand.

Thank you.
• Oct 24th 2010, 06:38 AM
HallsofIvy
It says that $\displaystyle G_a$, the isotropy group of a, is isomorphic to the quotient group $\displaystyle G/G_a$. Have you dealt with quotient groups?
• Oct 24th 2010, 06:59 AM
Sogan
Oh yes, thank you for your help.

Yes i know the Quotient Group. Now i'm clear in that point.

thanks a lot.