Abnormal subgroups in classical groups that correspond to closed root sets. (Q844492)

The article under review is devoted to the classification of minimal abnormal subgroups in Chevalley groups that contain a minimal split torus. Let \(\Phi\) be an irreducible root system, \(S\subseteq\Phi\) be a closed root subset. Denote by \(S_r:=S\cap-S\) the `reductive part' of \(S\) and by \(\omega(S)\) the action of an element \(\omega\) of the Weyl group \(W(S)\) on the root system \(S\), respectively. Denote also by \(S_\omega\) the subgroup generated by \(S\cup\omega(S)\). The authors show that the problem of the classification of minimal abnormal subgroups in Chevalley groups that contain a minimal split torus is reduced to the following problem on root systems. Problem. Describe closed root sets \(S\subseteq\Phi\) such that \(\omega\in W(S_\omega)=W(\langle S\cup\omega(S)\rangle^r)\) for any \(\omega\in W\). The main result obtained can be seen in the following Theorem. Let \(\Phi\) be a classical root system, i.e., of type \(A_n\), \(B_n\), \(C_n\) or \(D_n\), and let a closed root subset \(S\subseteq \Phi\) be given. Then the following conditions are equivalent: (i) there is \(\omega\in W(\Phi)\) such that \(\omega\not\in W(S_\omega)\); (ii) there is \(\omega\in W(\Phi)\) such that \(\omega\not\in W(S_\omega)\), \(\omega^2=1\).