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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..

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

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

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