Let be a Coxeter group and a reflection. If the group of order 2 generated by is the intersection of all the maximal finite subgroups of that contain it, then any isomorphism from to a Coxeter group must take to a reflection in . The aim of this paper is to show how to determine, by inspection of the Coxeter graph, the intersection of the maximal finite subgroups containing . In particular we show that the condition above is satisfied whenever is infinite and irreducible, and has the property that all rank two parabolic subgroups are finite. So in this case all isomorphisms map reflections to reflections.
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Bernhard Mühlherr, W. N. Franzsen, R. B. Howlett, Reflections in abstract Coxeter groups. Comment. Math. Helv. 81 (2006), no. 3, pp. 665–697DOI 10.4171/CMH/69