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The endomorphism rings of permutation modules of $\frac{3}{2}$-transitive permutation groups

Jiawei He, Xiaogang Li

Abstract

Recent classification of $\frac{3}{2}$-transitive permutation groups leaves us with six families of groups which are $2$-transitive, or Frobenius, or one-dimensional affine, or the affine solvable subgroups of $ \mathrm{AGL}(2, q)$, or special projective linear group $\mathrm{PSL}(2, q)$, or $\mathrm{PΓL}(2, q)$, where $q=2^p $ with $p$ prime. According to a case by case analysis, we prove that the endomorphism ring of the natural permutation module for a $\frac{3}{2}$-transitive permutation group is a symmetric algebra.

The endomorphism rings of permutation modules of $\frac{3}{2}$-transitive permutation groups

Abstract

Recent classification of -transitive permutation groups leaves us with six families of groups which are -transitive, or Frobenius, or one-dimensional affine, or the affine solvable subgroups of , or special projective linear group , or , where with prime. According to a case by case analysis, we prove that the endomorphism ring of the natural permutation module for a -transitive permutation group is a symmetric algebra.
Paper Structure (6 sections, 8 theorems, 36 equations)

This paper contains 6 sections, 8 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Symmetric Algebras
  4. Coherent Configurations
  5. Schur rings
  6. Proofs

Key Result

Theorem 1.1

Let $k$ be a field and $G$ be a $\frac{3}{2}$-transitive permutation groups on a set $\Omega$, then the endomorphism ring of the natural permutation module $k\Omega$ for $kG$ is a symmetric algebra.

Theorems & Definitions (14)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.1
  • Lemma 3.1
  • Example 3.1
  • Theorem 3.1
  • Lemma 3.2
  • ...and 4 more