Finite-dimensional pseudofinite groups of small dimension, without CFSG

TL;DR

This paper characterizes finite-dimensional pseudofinite groups of small dimension without relying on the classification of finite simple groups, extending understanding of their structure and classification.

Contribution

It provides a classification of pseudofinite finite-dimensional groups with small dimension, independent of CFSG, especially for dimension less than 4.

Findings

Pseudofinite groups of dimension less than 4 are classified as isomorphic to PSL(2,F) over pseudofinite fields.

The classification of these groups does not depend on the CFSG.

Finite-dimensional pseudofinite groups with small dimension exhibit specific structural properties.

Abstract

Any simple pseudofinite group G is known to be 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 G is finite-dimensional with additive and fine dimension and, in particular, that if dim(G)=3 then G is isomorphic to PSL(2,F) for some pseudofinite field F. We describe pseudofinite finite-dimensional groups when the dimension is fine, additive and \<4 and, in particular, show that the classification G isomorphic to PSL(2,F) is independent from CFSG.