Local commensurability graphs of solvable groups |
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Authors: | Khalid Bou-Rabee Chen Shi |
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Affiliation: | 1. School of Mathematics, CCNY CUNY, New York City, New York, USAkbourabee@ccny.cuny.edu;3. School of Mathematics, CCNY CUNY, New York City, New York, USA |
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Abstract: | The commensurability index between two subgroups A,B of a group G is [A:A∩B][B:A∩B]. This gives a notion of distance among finite index subgroups of G, which is encoded in the p-local commensurability graphs of G. We show that for any metabelian group, any component of the p-local commensurabilty graph of G has diameter bounded above by 4. However, no universal upper bound on diameters of components exists for the class of finite solvable groups. In the appendix we give a complete classification of components for upper triangular matrix groups in GL(2,𝔽q). |
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Keywords: | Borel subgroups commensurability containment graphs solvable groups |
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