Weyl group of $V_{4}$ in $S_{4}?$

Question

Can anyone please explain what a Weyl group is and what a Weyl group of $V_{4}$ in $S_{4}$ is? I do not understand this exercise.

Answer 1

I think I've read somewhere that the Weyl group of $H$ as a subgroup of $G$ is :

$$W_G(H):= N_G(H)/Z_G(H) $$

Where :

$$N_G(H):=\{g\in G\mid gHg^{-1}H\} $$

$$Z_G(H):=\{g\in G\mid \forall h\in H\text{, } ghg^{-1}=h\} $$

Some facts you need to do before understanding it :

1. $N_G(H)\leq G$.
2. $Z_G(H)\leq N_G(H)$.
3. $Z_G(H)\triangleleft N_G(H)$.

From this we see that $W_G(H)$ is a group.

4. There is a natural inclusion $W_G(H)\subseteq Aut(H)$.

What you need to do for your exercice :

5. $$N_{S_4}(V_4)=S_4 $$

6. $$Z_{S_4}(V_4)=V_4$$

7. Compute the automorphism group of $V_4$ and conclude.