Ultraproducts and Chevalley groups
type: Item
instance of: scholarly article
title: Ultraproducts and Chevalley groups
zbMATH Open document ID: 0921.03008
author: Françoise Point
published in: Archive for Mathematical Logic
publication date: 1998-09-08T00:00:00Z
review text: Given a simple non-trivial finite-dimensional Lie algebra \(L\), fields \(K_i\) and Chevalley groups \(L(K_i)\), we first prove that \(\prod_{\mathcal U}L(K_i)\) is isomorphic to \(L(\prod_{\mathcal U} K_i)\). Then we consider the case of Chevalley groups of twisted type \({}^nL\). We obtain a result analogous to the previous one. Given perfect fields \(K_i\) having the property that any element is either a square or the opposite of a square and Chevalley groups \({}^nL(K_i)\), then \(\prod_{\mathcal U}^nL(K_i)\) is isomorphic to \({}^nL(\prod_{\mathcal U} K_i)\). We apply our results to prove the decidability of the set of sentences true in almost all finite groups of the form \(L(K)\) where \(K\) is a finite field and \(L\) a fixed untwisted Chevalley type.
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 17B99
Mathematics Subject Classification ID: 20F10
zbMATH DE Number: 1186292
zbMATH Keywords: decidability
zbMATH Keywords: Lie algebra
zbMATH Keywords: Chevalley groups
MaRDI profile type: Publication
full work available at URL: s001530050131
OpenAlex ID: W2036011473
DOI: 10.1007/S001530050131
description: scientific article; zbMATH DE number 1186292