Ultraproducts and Chevalley groups

Abstract

Given a simple non-trivial finite-dimensional Lie algebra L, fields $K_i$ and Chevalley groups $L(K_i)$ , we first prove that $\Pi_{\mathcal{U}} L(K_i)$ is isomorphic to $L(\Pi_{\mathcal{U}}K_i)$ . Then we consider the case of Chevalley groups of twisted type ${}^n\!L$ . 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 ${}^n\!L(K_i)$ , then $\pu{}^n\!L(K_i)$ is isomorphic to ${}^n\!L(\pu K_i)$ . 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.