Normal subgroups of \(SL_{1,D}\) and the classification of finite simple groups
Let \(G\) be a simple simply connected algebraic group over an algebraic number field \(K\) and \(T\) the (finite) set of all nonarchimedean places \(v\) of \(K\) such that \(G\) is \(K_v\)-anisotropic. Define \(G(K,T)\) to be \(\prod_{v\in T}G(K_v)\) with the product topology if \(T\neq\emptyset\), and \(G(K,T)=\{e\}\) if \(T=\emptyset\). Let \(\delta\colon G(K)\to G(K,T)\) be the diagonal embedding in the first case, and the trivial homomorphism in the second case. \textit{V. P. Platonov} and \textit{A. S. Rapinchuk} [Algebraic Groups and Number Theory (Nauka, Moskwa, 1991; Zbl 0732.20027) (English translation: Pure Appl. Math. 139, Academic Press, 1993; Zbl 0841.20046)] conjecture that, for any noncentral normal subgroup \(N\subset G(K)\), there is an open normal subgroup \(W\subset G(K,T)\) such that \(N=\delta^{-1}(W)\). This conjecture has been proved for almost all \(K\)-isotropic groups and most \(K\)-anisotropic groups of type different from \(A_n\). So the major attention is now focussed on the case of anisotropic groups of type \(A_n\) and some results were obtained. In this paper, the authors prove that, for \(G=SL_{1,D}\) with \(D\) a central division algebra of degree 3 over \(K\), the Platonov-Rapinchuk conjecture holds. The proof is based on the classification of finite simple groups.
simple simply connected algebraic groups
scientific article; zbMATH DE number 1043986
scientific article