A new type of pronormal subgroups in Chevalley groups (Q5940190)

Let \(L\) be a simple finite-dimensional Lie algebra over an algebraically closed field of characteristic 0 with irreducible root system \(\Phi\), where not all roots have the same length. Let \(\Delta\) be a fixed system of simple roots and \(G(\Phi,F)\) the adjoint Chevalley group of type \(\Phi\) over a field \(F\) of characteristic other than 2, with \(G^l(\Phi,F)\) the subgroup generated by all \(x_r(t)\), \(r\in\Phi_l\), \(t\in F\), where \(\Phi_l\) is the set of long roots. The author's main result is that for \(\Phi\) other than of type \(G_2\), the subgroup \(G^l(\Phi,F)\) is `pronormal': that is, for any \(x\in G(\Phi,F)\) there is an element \(y\) in the subgroup generated by \(G^l(\Phi,F)\) and its conjugate by \(x\), with \(yTy^{-1}=xTx^{-1}\). Results of \textit{N. Vavilov} [in Lond. Math. Soc. Lect. Note Ser. 207, 233-280 (1995; Zbl 0879.20020)] establish the pronormality of the diagonal subgroup \(H\) of \(G(\Phi,F)\) and the unipotent subgroup \(U\) generated by \(x_r(t)\), \(t\in F\) for positive roots \(r\). This affords (Corollary 1.2) a characterization of the lattice of subgroups between \(G^l(\Phi,F)\) and \(G=G(\Phi,F)\) as the disjoint union of lattices of subgroups between \(E\) and \(N_{G^l(\Phi,F)}(E)\) for subgroups \(E\) generated by collections of conjugates of \(G^l(\Phi,F)\) in \(G\).