Apparaît dans la collection : Model Theory and Applications to Groups and Combinatorics / Théorie des modèles et applications en théorie des groupes et en combinatoire

A simple group is pseudofinite if and only if it is isomorphic to a (twisted) Chevalley group over a pseudofinite field. This celebrated result mostly follows from the work of Wilson in 1995 and heavily relies on the classification of finite simple groups (CFSG). It easily follows that any simple pseudofinite group $G$ is finite-dimensional. In particular, if $\operatorname{dim}(G)=3$ then $G$ is isomorphic to $\operatorname{PSL}(2, F)$ for some pseudofinite field $F$. In this talk, we describe the structures of finite-dimensional pseudofinite groups with dimension $<4$, without using CFSG. In the case $\operatorname{dim}(G)=3$ we show that either $G$ is soluble-by-finite or has a finite normal subgroup $Z$ so that $G / Z$ is a finite extension of $\operatorname{PSL}(2, F)$. This in particular implies that the classification $G \cong \operatorname{PSL}(2, F)$ from the above does not require CFSG. This is joint work with Frank Wagner.