{"title":"Finite-dimensional pseudofinite groups of small dimension, without CFSG","authors":"Ulla KarhumäkiAGL, Frank Olaf WagnerAGL","doi":"arxiv-2408.11484","DOIUrl":null,"url":null,"abstract":"Any simple pseudofinite group G is known to be isomorphic to a (twisted)\nChevalley group over a pseudofinite field. This celebrated result mostly\nfollows from the work of Wilson in 1995 and heavily relies on the\nclassification of finite simple groups (CFSG). It easily follows that G is\nfinite-dimensional with additive and fine dimension and, in particular, that if\ndim(G)=3 then G is isomorphic to PSL(2,F) for some pseudofinite field F. We\ndescribe pseudofinite finite-dimensional groups when the dimension is fine,\nadditive and \\<4 and, in particular, show that the classification G isomorphic\nto PSL(2,F) is independent from CFSG.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.11484","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.