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Let $(W,S)$ be a Coxeter system. For any $s,t\in S$, denote $m_{st}$ be the order of $st$. By a reflection, we mean the element in $W$ which conjugates to some simple reflection.

My question is: Let $\sigma,\tau$ be two reflections. Does the order of $\sigma\tau$ always divide some $m_{st}$ with $s,t\in S$ and $m_{st}<\infty$, or equal to $\infty$?

Expecting your reply, thank you.

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  • $\begingroup$ Yes: every finite reflection subgroup $W'$ with rank $r$ is a subgroup of a finite parabolic subgroup of rank $r$. So if $\sigma\tau$ has finite order, then $\langle \sigma,\tau\rangle$ is a reflection subgroup of the dihedral group of order $2m_{st}$ for some $s,t\in S$, so the order of $\sigma\tau$ divides $m_{st}$. $\endgroup$
    – Grant B.
    Commented Nov 11 at 5:47
  • $\begingroup$ Thank you for your reply $\endgroup$
    – user46809
    Commented Nov 11 at 8:48

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