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.