# 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

## Abstract

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