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- Title
IRREDUCIBLE COXETER GROUPS.
- Authors
PARIS, LUIS
- Abstract
We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that an indefinite irreducible Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. Let W be a Coxeter group. Write W = WX1 × ⋯ × WXb × WZ3, where WX1, ... , WXb are non-spherical irreducible Coxeter groups and WZ3 is a finite one. By a classical result, known as the Krull–Remak–Schmidt theorem, the group WZ3 has a decomposition WZ3 = H1 × ⋯ × Hq as a direct product of indecomposable groups, which is unique up to a central automorphism and a permutation of the factors. Now, W = WX1 × ⋯ × WXb × H1 × ⋯ × Hq is a decomposition of W as a direct product of indecomposable subgroups. We prove that such a decomposition is unique up to a central automorphism and a permutation of the factors. Write W = WX1 × ⋯ × WXa × WZ2 × WZ3, where WX1, ... , WXa are indefinite irreducible Coxeter groups, WZ2 is an affine Coxeter group whose irreducible components are all infinite, and WZ3 is a finite Coxeter group. The group WZ2 contains a finite index subgroup R isomorphic to ℤd, where d = |Z2| - b + a and b - a is the number of irreducible components of WZ2. Choose d copies R1, ... , Rd of ℤ such that R = R1 × ⋯ × Rd. Then G = WX1 × ⋯ × WXa × R1 × ⋯ × Rd is a virtual decomposition of W as a direct product of strongly indecomposable subgroups. We prove that such a virtual decomposition is unique up to commensurability and a permutation of the factors.
- Subjects
COXETER groups; GROUP theory; AUTOMORPHISMS; PERMUTATIONS; MATHEMATICAL decomposition
- Publication
International Journal of Algebra & Computation, 2007, Vol 17, Issue 3, p427
- ISSN
0218-1967
- Publication type
Article
- DOI
10.1142/S0218196707003779