Search results for "20F55"

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IRREDUCIBLE COXETER GROUPS

2004

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 dir…

[ MATH.MATH-GR ] Mathematics [math]/Group Theory [math.GR]General MathematicsGroup Theory (math.GR)0102 computer and information sciencesPoint group01 natural sciences[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR]CombinatoricsMathematics::Group TheoryFOS: Mathematics0101 mathematicsLongest element of a Coxeter groupMathematics::Representation Theory[MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR]MathematicsMathematics::CombinatoricsCoxeter notationMathematics::Rings and Algebras010102 general mathematicsCoxeter group010201 computation theory & mathematicsCoxeter complexArtin group20F55Indecomposable moduleMathematics - Group TheoryCoxeter elementInternational Journal of Algebra and Computation
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Automorphisms and abstract commensurators of 2-dimensional Artin groups

2004

In this paper we consider the class of 2-dimensional Artin groups with connected, large type, triangle-free defining graphs (type CLTTF). We classify these groups up to isomorphism, and describe a generating set for the automorphism group of each such Artin group. In the case where the defining graph has no separating edge or vertex we show that the Artin group is not abstractly commensurable to any other CLTTF Artin group. If, moreover, the defining graph satisfies a further `vertex rigidity' condition, then the abstract commensurator group of the Artin group is isomorphic to its automorphism group and generated by inner automorphisms, graph automorphisms (induced from automorphisms of the…

Vertex (graph theory)20F67CommensuratorCoxeter groupCoxeter group20F36InverseGroup Theory (math.GR)Automorphism2–dimensional Artin group20F36 20F55 20F65 20F67CombinatoricsMathematics::Group Theorytriangle freeGenerating set of a groupFOS: Mathematicscommensurator groupArtin groupGeometry and TopologyIsomorphism20F5520F65graph automorphismsMathematics - Group TheoryMathematics
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