Small groups of finite Morley rank with a supertight automorphism

TL;DR

This paper classifies certain infinite simple groups of finite Morley rank with a supertight automorphism, showing they are isomorphic to PGL_2 over an algebraically closed field of characteristic not 2.

Contribution

It provides a classification result for groups of finite Morley rank with a supertight automorphism, linking them to classical algebraic groups.

Findings

G is isomorphic to PGL_2(K) for some algebraically closed field K

The fixed-point subgroups are pseudofinite for all positive powers of the automorphism

The group has Pr"{u}fer 2-rank 1 and admits a supertight automorphism

Abstract

Let $G$ be an infinite simple group of finite Morley rank and of Pr\"{u}fer $2$-rank $1$ which admits a supertight automorphism $\alpha$ such that the fixed-point subgroup $C_G(\alpha^n)$ is pseudofinite for all integers $n > 0$. We prove that $G \cong {\rm PGL}_2(K)$ for some algebraically closed field $K$ of characteristic $\neq 2$.