The following exercise comes from section 2.11 in Humphreys' book Reflection groups and Coxeter Groups:
Use the method of this section to derive again the orders of the groups of types $A_n$, $B_n$, and $C_n$.
I have only thought about the type $A_n$ case so far. Depending on how this posts go, I might make a separate one for types $B_n$ and $C_n$.
In section 2.11, Humphreys uses the orbit-stabilizer theorem to deduce the order of the reflection groups of types $F_4$, $E_6$, $E_7$, and $E_8$. In each case, he use the "highest" root $\tilde{\alpha}$ of the root system associated to the group and looks at the stabilizer of it. In each case, he argues that $\tilde{\alpha}$ is orthogonal to all the simple roots except one, and states that the stabilizer/isotropy group must be of a certain type. E.g., in the $F_4$ case, $\tilde{\alpha}$ is orthogonal to every simple root except $\alpha_1$, so its stabilizer group is of type $C_3$ (honestly, I don't see why that's the case but whatever). And then the number of long roots gives the cardinality of the orbit of the highest root, although I don't see why that's the case either. From here, it's easy to work out the order of the group from the orbit-stabilizer theorem, modulo those two steps I don't understand.
So, let $W$ be the reflection the group of type $A_n$. Then $W = S_{n+1}$ which acts naturally acts on $V = \Bbb{R}^{n+1}$ by permuting the standard basis $\{e_i\}_{i=1}^{n+1}$ and then extending this action linearly to all of $V$. The root system is given by
$$\Phi = \{(x_1,...,x_{n+1}) \in V : ||(x_1,...,x_{n+1}||^2 = 2, \sum x_i = 0\} \cap \text{span}_{\Bbb{Z}} \{e_1,....,e_{n+1}\}$$
$$= \{e_i-e_j : 1 \le i \neq j \le n+1\},$$ which consists of $n(n+1)$ vectors, the simple roots are
$$\Delta = \{\alpha_1 = e_1 - e_2,..., \alpha_n = e_n - e_{n+1}\},$$
and $\tilde{\alpha} = e_1 - e_{n+1}$ is the highest root. It's clear that $\tilde{\alpha}$ is orthogonal to every simple root except $\alpha_1$ and $\alpha_n$...But what does this tell us about the type of the stabilizer subgroup. In the type $A_n$ case (or any case for that matter), I don't know the number of long roots.
I could use some help working out the all three cases.