Irreducible Coxeter Groups

We prove that an infinite irreducible Coxeter group cannot be a non-trivial direct product. Let W be a Coxeter group, and write W = W1 × · · · ×Wp ×Wp+1, where W1, . . . ,Wp are infinite irreducible Coxeter groups, and Wp+1 is a finite one. As an application of the main result, we obtain that W1, . . . ,Wp are unique and Wp+1 is unique up to isomorphism. That is, if W = W̃1 × · · · × W̃q × W̃q+1 is another decomposition where W̃1, . . . , W̃q are infinite irreducible Coxeter groups and W̃q+1 is a finite one, then p = q, W̃p+1 ≃ Wp+1, and there is a permutation χ ∈ Symp such that W̃χ(i) = Wi for all 1 ≤ i ≤ p. AMS Subject Classification: Primary 20F55.