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arXiv · 1908.00613

Finite Permutation Groups with Few Orbits Under the Action on the Power Set

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

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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Alexander Betz, Max Chao-Haft, Ting Gong, Thomas Michael Keller, Anthony Ter-Saakov, Yong Yang. 2019-08-01. Finite Permutation Groups with Few Orbits Under the Action on the Power Set. https://arxiv.org/abs/1908.00613

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