Groups contained between groups of Lie type over various fields
Let \(G(R)\) be a Chevalley group over a field \(R\) and let \(K/R\) be an algebraic field extension. The author considers groups \(M\) such that \(G(R)\subset M\subset G(K)\). For more details assume that \(| R|>1+\max(r,r)/(s,s)\) where \(r,s\) are simple roots, as well as the rank of \(G\) is \(>1\) (a misprint: it is typed \(G>1\)). Assume also that \(R\) is perfect if \(1<char R<\max(r,r)/(s,s)\). Theorem 1 asserts that \(G(R)\subset M\subset G(K)\) implies \(G(P)\subset M\subset\widehat G(P)\) under these assumptions where \(P\) is a field and \(R\subset P\subset K\) and \(\widehat G(P)\) is an extended Chevalley group, that is the group \(G(R)\) extended by diagonal automorphisms. Actually Theorem 1 is valid under the weaker assumption \(M\cap x_ r(R)\neq 0\) for any root \(r\) instead of \(M\supset G(R)\). Moreover the author proves a certain version of this theorem for twisted forms of Chevalley groups.
scientific article; zbMATH DE number 3865578