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The Chain Group of a Forest
For every labeled forest \(\mathsf{F}\) with set of vertices \([n]\) we can consider the subgroup \(G\) of the symmetric group \(S_n\) that is generated by all the cycles determined by all maximal paths of \(\mathsf{F}\). We say that \(G\) is the chain group of the forest \(\mathsf{F}\). In this pap...
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Published in: | arXiv.org 2017-06 |
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Main Authors: | , |
Format: | Article |
Language: | English |
Subjects: | |
Online Access: | Get full text |
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Summary: | For every labeled forest \(\mathsf{F}\) with set of vertices \([n]\) we can consider the subgroup \(G\) of the symmetric group \(S_n\) that is generated by all the cycles determined by all maximal paths of \(\mathsf{F}\). We say that \(G\) is the chain group of the forest \(\mathsf{F}\). In this paper we study the relation between a forest and its chain group. In particular, we find the chain groups of the members of several families of forests. Finally, we prove that no copy of the dihedral group of cardinality \(2n\) inside \(S_n\) can be achieved as the chain group of any forest. |
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ISSN: | 2331-8422 |