Ultraproducts and Chevalley groups
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.
scientific article; zbMATH DE number 1186292