Fixed point subgroups of a supertight automorphism

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 ∈ N ∖ { 0 } . It is know (using CFSG) that the socle S n : = Soc ( P n ) is a (twisted) Chevalley group over a pseudofinite field. We prove that there is r ∈ N ∖ { 0 } so that for each n we have [ P n : S n ] < r and that there is no m ∈ N ∖ { 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 pr 2 ( G ) = 1 , the use of CFSG is not needed.