Reflections in abstract Coxeter groups

  • Bernhard Mühlherr

    Universität Gießen, Giessen, Germany
  • W. N. Franzsen

    Australian Catholic University, Strathfield, Australia
  • R. B. Howlett

    The University of Sydney, Australia


Let WW be a Coxeter group and rWr\in W a reflection. If the group of order 2 generated by rr is the intersection of all the maximal finite subgroups of WW that contain it, then any isomorphism from WW to a Coxeter group WW' must take rr to a reflection in WW'. 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 rr. In particular we show that the condition above is satisfied whenever WW 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.

Cite this article

Bernhard Mühlherr, W. N. Franzsen, R. B. Howlett, Reflections in abstract Coxeter groups. Comment. Math. Helv. 81 (2006), no. 3, pp. 665–697

DOI 10.4171/CMH/69