I have just completed Exercise 1.2 in the book "Combinatorics of Coxeter Groups" stated below:
Show that there exist Coxeter systems $(W,S)$ and $(W',S')$ with $|S|\neq|S'|$ such that $W\cong W'$ as abstract groups.
[Hint: Consider the dihedral group $D_6$ of order 12]
I wonder if this problem could be extended to finite dihedral groups $D_n$ of order $2n$, or even the infinite dihedral group $D_\infty$. Could anyone give me some insights on this? Thanks!
