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Finite Permutation Groups with Few Orbits Under the Action on the Power Set
We study the orbits under the natural action of a permutation group \(G \subseteq S_n\) on the powerset \(\mathscr{P}(\{1, \dots , n\})\). The permutation groups having exactly \(n+1\) orbits on the powerset can be characterized as set-transitive groups and were fully classified in \cite{BP55}. In t...
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Published in: | arXiv.org 2021-08 |
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Main Authors: | , , , , , |
Format: | Article |
Language: | English |
Subjects: | |
Online Access: | Get full text |
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Summary: | We study the orbits under the natural action of a permutation group \(G \subseteq S_n\) on the powerset \(\mathscr{P}(\{1, \dots , n\})\). The permutation groups having exactly \(n+1\) orbits on the powerset can be characterized as set-transitive groups and were fully classified in \cite{BP55}. In this paper, we establish a general method that allows one to classify the permutation groups with \(n+r\) set-orbits for a given \(r\), and apply it to integers \(2 \leq r \leq 15\) using the computer algebra system GAP. |
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ISSN: | 2331-8422 |