arXiv subjects
Publications and source records attributed to Pınar Uğurlu.
Let $G$ be an infinite simple group of finite Morley rank and of Prüfer $2$-rank $1$ which admits a supertight automorphism $α$ such that the fixed-point subgroup $C_G(α^n)$ is pseudofinite for all integers $n > 0$. We prove that $G \cong {\rm PGL}_2(K)$ for some algebraically closed field $K$ of characteristic $\neq 2$.
We prove the conjugacy of Sylow $p$-subgroups of linear pseudofinite groups under the assumption of the existence of a finite Sylow $p$-subgroup. We also give an example of a linear pseudofinite group with non-conjugate Sylow $2$-subgroups.
We prove the conjugacy of Sylow $2$-subgroups in pseudofinite $\mathfrak{M}_c$ (in particular linear) groups under the assumption that there is at least one finite Sylow $2$-subgroup. We observe the importance of the pseudofiniteness assumption by analyzing an example of a linear group with non-conjugate finite Sylow $2$-subgroups which was constructed by Platonov.
We prove that if the group of fixed points of a generic automorphism of a simple group of finite Morley rank is pseudofinite, then this group is an extension of a (twisted) Chevalley group over a pseudofinite field. On the way to obtain this result, we classify non-abelian definably simple pseudofinite groups of finite centralizer dimension (using the ideas of John S. Wilson).