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Fixed point subgroups of a supertight automorphism

Ulla Karhumäki

Abstract

Let $G$ be an infinite simple group of finite Morley rank and $α$ a supertight automorphism of $G$ so that the fixed point subgroup $P_n:=C_G(α^n)$ is pseudofinite for all $n\in \mathbb{N}\setminus\{0\}$. It is know (using CFSG) that the socle $S_n:={\rm Soc}(P_n)$ is a (twisted) Chevalley group over a pseudofinite field. We prove that there is $r\in \mathbb{N}\setminus\{0\}$ so that for each $n$ we have $[P_n:S_n] < r$ and that there is no $m \in \mathbb{N}\setminus \{0\}$ so that for each $n$ the sizes of the Sylow $2$-subgroups of $S_n$ are bounded by $m$. We also note that in the recent identification result of $G$ under the assumption ${\rm pr}_2(G)=1$, the use of CFSG is not needed.

Fixed point subgroups of a supertight automorphism

Abstract

Let be an infinite simple group of finite Morley rank and a supertight automorphism of so that the fixed point subgroup is pseudofinite for all . It is know (using CFSG) that the socle is a (twisted) Chevalley group over a pseudofinite field. We prove that there is so that for each we have and that there is no so that for each the sizes of the Sylow -subgroups of are bounded by . We also note that in the recent identification result of under the assumption , the use of CFSG is not needed.
Paper Structure (11 sections, 9 theorems, 6 equations)

This paper contains 11 sections, 9 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chevalley groups
  4. Pseudofinite groups
  5. Groups of finite Morley rank
  6. Fixed point subgroups of supertight automorphisms
  7. Pseudofinite fixed point subgroups
  8. Killing the field automorphisms
  9. Growing the Sylow $2$-subgroups
  10. Proof of the main theorem
  11. The role of CFSG in the supertight programme

Key Result

Theorem 1.1

Let $G$ be a simple group of finite Morley rank with a supertight automorphism $\alpha$ whose fixed point subgroup $P_n:=C_G(\alpha^n)$ is pseudofinite for all $n\in \mathbb{N}\setminus \{0\}$. Set $S_n:={\rm Soc}(P_n)$. Then the following holds.

Theorems & Definitions (20)

  • Definition
  • Theorem 1.1
  • Proposition 1.2
  • Definition 2.1
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Example 1
  • ...and 10 more