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Betz, Alexander; Chao-Haft, Max; Gong, Ting; Keller, Thomas Michael; Ter-Saakov, Anthony; Yang, Yong

Finite permutation groups with few orbits under the action on the power set. (English) Zbl 1508.20003

Rocky Mt. J. Math. 51, No. 5, 1553-1565 (2021).
Summary: We study the orbits under the natural action of a permutation group \(G \leq \mathsf{S}_n\) on the powerset \(\mathcal{P}(\{1, \ldots, n\})\). The permutation groups having exactly \(n+1\) orbits on the powerset can be characterized as set-transitive groups and were fully classified by R. A. Beaumont and R. P. Peterson [Can. J. Math. 7, 35–42 (1955; Zbl 0064.02504)]. 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 \(2 \leq r \leq 15\) with the help of GAP.

Cited in 1 Review

MSC:

20B05 General theory for finite permutation groups

Keywords:

permutation groups; power set; orbit

Citations:

Zbl 0064.02504

Software:

GAP
PDFBibTeX XMLCite
\textit{A. Betz} et al., Rocky Mt. J. Math. 51, No. 5, 1553--1565 (2021; Zbl 1508.20003)
Full Text: DOI arXiv Link

References:

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