ON OUTER AUTOMORPHISM-GROUPS OF COXETER GROUPS

It is shown that the outer automorphism group of a Coxeter group W of finite rank is finite if the Coxeter graph contains no infinite bonds. A key step in the proof is to show that if the group is irreducible a nd Pi(1) and Pi(2) any two bases of the root system of W, then Pi(2) = +/-w Pi(1) for some w is an element of W. The proof of this latter fa ct employs some properties of the dominance order on the root system i ntroduced by Brink and Howlett.