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Classifying Primitive Solvable Permutation Groups of Rank 5 and 6

Anakin Dey, Kolton O'Neal, Duc Van Khanh Tran, Camron Upshur, Yong Yang

Abstract

Let $G$ be a finite solvable permutation group acting faithfully and primitively on a finite set $Ω$. Let $G_0$ be the stabilizer of a point $α\in Ω$ The rank of $G$ is defined as the number of orbits of $G_0$ in $Ω$, including the trivial orbit $\{α\}$. In this paper, we completely classify the cases where $G$ has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.

Classifying Primitive Solvable Permutation Groups of Rank 5 and 6

Abstract

Let be a finite solvable permutation group acting faithfully and primitively on a finite set . Let be the stabilizer of a point The rank of is defined as the number of orbits of in , including the trivial orbit . In this paper, we completely classify the cases where has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.
Paper Structure (5 sections, 13 theorems, 15 equations, 5 tables)

This paper contains 5 sections, 13 theorems, 15 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Computation
  4. Results
  5. Future Work

Key Result

Theorem 1.1

Suppose $G = V \rtimes G_0$ is a finite primitive solvable permutation group of rank at most 6, where $G_0$ acts on $V$ as an irreducible subgroup of $\mathop{\mathrm{GL}}\nolimits(V)$. At least one of the following is true:

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 2.1: Theorem 2.2 of reg_orbit_1, Theorem 2.2 of reg_orbit_2, and Theorem 2.1 of reg_orbit_3
  • Lemma 2.2: Lemma 2.3 of reg_orbit_2
  • Proposition 2.3
  • Proof
  • Lemma 2.4
  • Proof
  • Lemma 2.5
  • Proof
  • Lemma 2.6
  • ...and 15 more