# Orbits, stabilizers, and fixed points

Question : Let $H$ be a subgroup of $G$ and let $H$ act on $G/H$ by translation. Find the orbits, stabilizers, and fixed points of the action.

I think stabilizer is g$H$g^-1 ...but I don't know orbit and fixed points..

• ...please...? And what have you done so far?! – DonAntonio Apr 18 '13 at 2:05
• I don't want to. (It's not polite to phrase your questions in the form of commands.) – Qiaochu Yuan Apr 18 '13 at 2:18
• sorry..I don't Know how can I approach this..please help me – Yoonju Kim Apr 18 '13 at 2:25

Hints:

The action is

$$\forall\,x,g\in G\;,\;\;x\cdot(gH):=(xg)H$$

letting $\,X\,$ be the set of all left cosets, thus:

$$\mathcal Orb(xH):=\{\,gxH\;;\;g\in G\,\}=X$$

Can you see why is there only one orbit (i.e., the action is transitive)?

$$Stab(xH):=\{\,g\in G\;;\;gxH=xH\iff x^{-1}gx\in H\,\}$$

Can you show, for example, that $\;Stab(H)=H\;$ ?

Fixed points:

$$xH\in X^G:=\{\,x\in X\;;\;gx=x\,\,\,\forall\,g\in G\}\iff \forall\,g\in G\;,\;\;gxH=xH\ldots$$