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 $0 \le i <n-1$ 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.

  • 5
    $\begingroup$ 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. $\endgroup$
    – YCor
    Aug 28, 2019 at 13:06
  • $\begingroup$ (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.) $\endgroup$
    – YCor
    Aug 28, 2019 at 13:59
  • 2
    $\begingroup$ Here you find a reference (including not just type $B_n$, but all irreducble finite Coxeter groups): cambridge.org/core/journals/… $\endgroup$
    – P. Wegener
    Aug 29, 2019 at 15:46


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