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

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

Group Theory 2021-08-03 v4

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

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.

Keywords

permutation groups nilpotent orbits permutations

Cite

@article{arxiv.1908.00613,
  title  = {Finite Permutation Groups with Few Orbits Under the Action on the Power Set},
  author = {Alexander Betz and Max Chao-Haft and Ting Gong and Thomas Michael Keller and Anthony Ter-Saakov and Yong Yang},
  journal= {arXiv preprint arXiv:1908.00613},
  year   = {2021}
}

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

Abstract

Keywords

Cite

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