On the isomorphism problem, indecomposability and the automorphism groups of Coxeter groups

Let (W, S) and (W′, S′) be Coxeter systems, {W _λ} _λ∈Λ and {W _{Λ ′}′ } _{λ′∈Λ ′} the sets of the irreducible components of W relative to S and of W′ relative to S′ respectively, and let f : W → W′ be an isomorphism of abstract groups. Their Coxeter graphs may not be isomorphic. We show that f(Π _{λ∈Λ,|Wλ|<∞} W _λ) = Π _{λ∈Λ,|Wλ′ ′|<∞}W _λ′′, and that there is a unique bijection φ: {λ ∈ Λ | |W _λ| = ∞} → {λ ′ ∈ Λ ′ | |W _λ′′| = ∞} such that f(W _λ) ≡ W _φ(λ)′ mod Z(W′) for every λ ∈ Λ with |W _λ| = ∞, where Z(W′) is the center of W0. We also determine which two finite Coxeter groups are isomorphic. Our result reduces the problem of deciding whether two Coxeter groups are isomorphic to the case of infinite irreducible Coxeter groups. As a corollary we determine which irreducible Coxeter group is directly indecomposable as an abstract group. In particular, any infinite irreducible Coxeter group is directly indecomposable.