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Permutation groups of prime power degree and $p$-complements

Gareth A. Jones, Sezgin Sezer

Abstract

Extending earlier work of Guralnick and of Cai and Zhang, we classify the almost simple groups which have transitive permutation representations of prime power degree $p^k$, and those which have $p$-complements (stabilisers of order coprime to $p$ in such representations). We deduce that every primitive permutation group of prime power degree has a regular subgroup, and that any two faithful primitive representations of a group, of the same prime power degree, are equivalent under automorphisms. In general, $p$-complements in a finite group can be inequivalent under automorphisms, or even non-isomorphic. We extend examples of such phenomena due to Buturlakin, Revin and Nesterov by showing that the number of inequivalent classes of complements can be arbitrarily large. Questions concerning the existence of prime power representations and $p$-complements in groups with socle ${\rm PSL}_d(q)$ are related to some difficult open problems in Number Theory.

Permutation groups of prime power degree and $p$-complements

Abstract

Extending earlier work of Guralnick and of Cai and Zhang, we classify the almost simple groups which have transitive permutation representations of prime power degree , and those which have -complements (stabilisers of order coprime to in such representations). We deduce that every primitive permutation group of prime power degree has a regular subgroup, and that any two faithful primitive representations of a group, of the same prime power degree, are equivalent under automorphisms. In general, -complements in a finite group can be inequivalent under automorphisms, or even non-isomorphic. We extend examples of such phenomena due to Buturlakin, Revin and Nesterov by showing that the number of inequivalent classes of complements can be arbitrarily large. Questions concerning the existence of prime power representations and -complements in groups with socle are related to some difficult open problems in Number Theory.
Paper Structure (15 sections, 23 theorems, 21 equations, 1 table)

This paper contains 15 sections, 23 theorems, 21 equations, 1 table.

Table of Contents

  1. Introduction
  2. $p$-complements
  3. Conjugacy classes of $p$-complements
  4. $p$-complements and permutation groups
  5. Permutation groups of prime power degree
  6. Simple groups
  7. $p$-complements in ${\rm PSL}_d(q)$
  8. Almost simple groups
  9. Almost simple groups in case (B)
  10. Extension to all primitive groups of prime power degree
  11. Proof of Corollary \ref{['cor:regsgp']}
  12. Proof of Corollary \ref{['cor:autequiv']}
  13. Conjectures from Number Theory
  14. The Bateman--Horn Conjecture
  15. The Nagell--Ljunggren Conjecture

Key Result

Theorem 1.1

The almost simple groups which are transitive permutation groups of prime power degree $n=p^k$ are as follows: There are two such permutation representations for $G$ in (B4) and (B5), and for ${\rm PSL}_2(11)$ in (C), transposed by an outer automorphism of $G$ in each case. In all other cases each group has just one such representation. All of these groups except those in (A2) are primitive, and

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1: Nesterov
  • Proposition 4.1
  • Lemma 4.2
  • Corollary 4.3
  • ...and 13 more