Intepretation of the fact that $\operatorname{stab}_G(g \cdot x) = g\operatorname{stab}_G(x)g^{-1}$

Question

I know how to prove the fact that $\operatorname{stab}_G(g \cdot x) = g\operatorname{stab}_G(x)g^{-1}$ but I'm having trouble finding meaning behind it. Does someone know why this is useful or has an intuitive view of this matter?

Answer 1

Here's an intuition.

Suppose you are standing on the point $g \cdot x$.

Now three things happen quickly in succession:

1. $g^{-1}$ moves you from $g \cdot x$ to $x$;
2. Someone spins the space around $x$ but leaves you fixed in position at $x$ (although perhaps leaving you somewhat dizzy)
3. And then $g$ moves you back from $x$ to $g\cdot x$.

The result: You started standing at $g \cdot x$, and you're still standing at $g \cdot x$.