# What are the normal subgroups of the finite Coxeter Groups of type Bn?

Let $$B_n = \langle \rho_0,\rho_1,\ldots,\rho_{n-1} \rangle$$ subject to the relations that $$(\rho_i\rho_j)^{m_{i,j}} = id$$ with $$m_{i,i} = 1$$, $$m_{i,j} = 2$$ for $$|i-j|\ge 2$$, $$m_{i,i+1} =3$$ for $$0 \le i and finally $$m_{n-1,n} =4$$. What are the normal subgroups of $$B_n$$ and where might I find a reference? Computationally, I see that for all $$n>4$$ there are 9 of them and I would like to find some source which describes them explicitly.

• I don't know a reference, but it's an exercise to determine them. This is the wreath product $C_2\wr_n S_n$. The normal subgroups contained in $C_2^n$ are $4$ many: $0$, the constants (of order $2$), the sum 0 subgroup $(C_2^n)_0$ (of index $2$) and the whole $C_2^n$. Others intersect $C_2^n$ at least in the sum $0$ subgroup. If the intersection is $C_2^n$, we get the 2 nontrivial normal subgroups of $S_n$. If the intersection is $(C_2^n)_0$: the quotient by the latter is $S_n\times C_2$. Its nontrivial normal subgroups not containing $C_2$ are $A_n$ and $S_n$. This gives $9$ normal subgroups.
– YCor
Aug 28, 2019 at 13:06
• (I used $n\ge 5$ in using the classification of normal subgroups of $S_n$. Also $n\ge 3$ is used at some point because I used that the action modulo constants is non-trivial.)
– YCor
Aug 28, 2019 at 13:59
• Here you find a reference (including not just type $B_n$, but all irreducble finite Coxeter groups): cambridge.org/core/journals/… Aug 29, 2019 at 15:46