# Orbit and stabilisers of subgroups of S4

We are given that for a group $$G$$ and set $$S$$ denoting the set of subgroups of $$G$$:

$$\rho(g,H) =gHg^{-1}$$ where $$g \in G, H \leqslant G$$ defines a left action of G on S.

We are then asked to determine $$Orb(H)$$ and $$Stab(H)$$ for the following subgroups of $$G=S_4$$:

$$H=V_4$$, $$H=Sym\{1,2,3\}$$ and $$H=\langle (1234) \rangle$$.

Any help would be appreciated. For $$H=V_4$$, I thought $$Orb(V_4)= V_4$$ and $$Stab(V_4)=S_4$$ but this clearly doesn't agree with the Orbit-Stabiliser theorem so not too sure where I've gone wrong.

• Isn't the orbit-stabilizer theorem for one element? – J. W. Tanner May 15 '20 at 17:07
• @J.W.Tanner an element of the set S. here S is the set of subgroups and so V4 is an element of the set. – DietCola01 May 15 '20 at 17:09
• Ya, sorry, I had missed that, it is clearly stated. I have deleted my previous comment. – user750041 May 15 '20 at 17:16

For $$H=V_4$$, I thought $$\mathrm{Orb}(V_4)=\{V_4\}$$ and $$\mathrm{Stab}(V_4)=S_4$$ but this clearly doesn't agree with the Orbit-Stabiliser theorem so not too sure where I've gone wrong.
You're wrong about being wrong. There is no disagreement with the orbit-stabizer theorem. Since the stabilizer is the whole group, the size of the quotient $$G/\mathrm{Stab}$$ is $$1$$, which is the size of the orbit $$\{V_4\}$$ (notice we must put $$V_4$$ in parentheses, since it is a single element of the orbit).
Any ideas for $$H=S_3$$ and $$H=\langle (1234)\rangle$$?
• Oh I see thank you! For $H=S_3$ I was thinking it may have something to do with $S_4/V_4 \cong S_3$ (this was shown earlier on) but not too sure from there and the last one again i'm not too sure – DietCola01 May 15 '20 at 18:46
• @DietCola01 That's a good fact to know, but not relevant at the moment. Since you're working with small groups, you can get your hands dirty by working out concrete examples. For instance if $g=(1234)$ and $H=\mathrm{Sym}(\{1,2,3\})$ then what are the elements of $gHg^{-1}$; can you describe $gHg^{-1}$ any way other than listing its elements? – runway44 May 15 '20 at 18:55