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 $\operatorname{PSL}_2(F)$ for some pseudofinite field $F$. We describe pseudofinite groups of fine and additive dimension $\leqslant 3$ and, in particular, show that the classification $G \cong \operatorname{PSL}_2(F)$ is independent of CFSG.
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Keywords: pseudofinite groups, finite-dimensional groups, classification of finite simple groups, simple groups of Lie type
Ulla Karhumäki 1 ; Frank O. Wagner 2
CC-BY-NC-ND 4.0
@article{CML_2025__17__55_0,
author = {Ulla Karhum\"aki and Frank O. Wagner},
title = {Finite-dimensional pseudofinite groups of small dimension, without {CFSG}},
journal = {Confluentes Mathematici},
pages = {55--71},
year = {2025},
publisher = {Institut Camille Jordan},
volume = {17},
doi = {10.5802/cml.100},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.5802/cml.100/}
}
TY - JOUR AU - Ulla Karhumäki AU - Frank O. Wagner TI - Finite-dimensional pseudofinite groups of small dimension, without CFSG JO - Confluentes Mathematici PY - 2025 SP - 55 EP - 71 VL - 17 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.100/ DO - 10.5802/cml.100 LA - en ID - CML_2025__17__55_0 ER -
%0 Journal Article %A Ulla Karhumäki %A Frank O. Wagner %T Finite-dimensional pseudofinite groups of small dimension, without CFSG %J Confluentes Mathematici %D 2025 %P 55-71 %V 17 %I Institut Camille Jordan %U https://cml.centre-mersenne.org/articles/10.5802/cml.100/ %R 10.5802/cml.100 %G en %F CML_2025__17__55_0
Ulla Karhumäki; Frank O. Wagner. Finite-dimensional pseudofinite groups of small dimension, without CFSG. Confluentes Mathematici, Tome 17 (2025), pp. 55-71. doi: 10.5802/cml.100
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