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The proportion of derangements characterizes the symmetric and alternating groups

Bjorn Poonen, Kaloyan Slavov

Abstract

Let $G$ be a subgroup of the symmetric group $S_n$. If the proportion of fixed-point-free elements in $G$ (or a coset) equals the proportion of fixed-point-free elements in $S_n$, then $G=S_n$. The analogue for $A_n$ holds if $n \ge 7$. We give an application to monodromy groups.

The proportion of derangements characterizes the symmetric and alternating groups

Abstract

Let be a subgroup of the symmetric group . If the proportion of fixed-point-free elements in (or a coset) equals the proportion of fixed-point-free elements in , then . The analogue for holds if . We give an application to monodromy groups.

Paper Structure

This paper contains 11 sections, 7 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Derangements in permutation groups
  3. Application to monodromy
  4. Structure of the paper
  5. Primitive permutation groups
  6. Imprimitive but transitive permutation groups
  7. Intransitive permutation groups
  8. Alternating group
  9. Primitive permutation groups
  10. Imprimitive permutation groups that preserve a partition into blocks of equal size
  11. Intransitive subgroups

Key Result

Theorem \oldthetheorem

Let $G$ be a subgroup of the symmetric group $S_n$ for some $n \ge 1$. Let $C$ be a coset of $G$ in $S_n$. If then $G=C=S_n$.

Theorems & Definitions (16)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 6 more