# Subgroups of finite reflection groups that do not fix a point

Let $(W,S)$ be a finite irreducible Coxeter-System of rank $n$ and $E$ be a real reflection representation of $W$. Let $x\in E$ and suppose that the isotropy group of $x$ is generated by one element in $S$. Now which are the subgroups of rank $n-1$ that do not stabilize $x$?

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When you say "do not stabilize" do you mean that the stabilizer of $x$ in the subgroup is trivial? –  S. Carnahan Nov 2 '13 at 0:42
yes, that is what I mean –  Tom Nov 4 '13 at 9:01
If $s_1$ to $s_n$ are the roots and $s_1$ is the root which fixes $x$ then the group generated by $s_2$ to $s_n$ would be what I need. However, is this all that can happen? –  Tom Nov 4 '13 at 12:38
Tom: There are other possibilities too. Consider do instance what happens in rank 2 cases. –  Misha Nov 4 '13 at 19:39
No I am a bit confused specially by the "answer" of Ben. –  Tom Nov 5 '13 at 8:22