Reference: p. 8 http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/gpaction.pdf
This PDF doesn't unfold all the steps hence can someone please notify me of bungles? Thank you.
I tried: $G$ acts on the left cosets of $H$ by left multiplication, so $ g \cdot xH = gxH $
(1.) $\mathrm{orb}_{xH} := \{g \cdot xH : g\in G\} = {\{gxH : g \in G\}}.$ $\because g,x \in G \therefore gx \in G \text{ hence } = \{(gx)H : g \in G\} = G/H .$
I understand $ \color{red}{\mathrm{Orb}_{\{H\}} := \{g \cdot \{H\} : g \in G\}} $. But how is my work overhead?
3.$\text{}$ How does $\mathrm{Stab}_{aH} = \{g : gaH = aH \} = \{g : a^{-1}ga \in H\} = \color{blue}{aHa^{-1}} $
I understand: $ \color{blue}{aHa^{-1} = \{h \in H : aha^{-1}\}}$
4.$\text{}$ How do you determine the fixed points craftily? I know the definition for $x$ to be a fixed point: $g \cdot x = x \; \forall g \in G$. Do I solve for $x$?
To boot, I tried from (3.) $g \cdot xH = xH \iff x^{-1}gx \in H \iff x \in H \text{ and } g \in H \iff G = H. $
But $g \cdot xH = xH $ isn't the definition of a fixed point?