The <Coxeter number> is the order of a <Coxeter element>. In the stated families the values are
$$
\begin{array}{c|c|c}
W&\text{Coxeter type}&h\\ \hline
D_{2n}&I_2(n)&n\\
S_n&A_{n-1}&n\\
SS_n&B_n&2n\\
ESS_n&D_n&2n-2.
\end{array}
$$
Here $D_{2n}$ is the <dihedral group> of order $2n$, $S_n$ is the <symmetric group>, $SS_n$ is the <signed symmetric group>, and $ESS_n$ is the <even signed symmetric group>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 4 / a / Solution

{title2=$ESS_n$}
{wiki=Coxeter_group#Finite_Coxeter_groups}

The even signed symmetric group is the index-two subgroup of signed permutations with an even number of sign changes. It is the Coxeter group of type $D_n$.


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 Incoming links: Even signed symmetric group