Garlands containing the general linear group over an intermediate field (Q1303325)

Let \(G\) be a group and \(T\) a subgroup of \(G\). Let \(\text{Lat}(T,G)\) denote the lattice of all intermediate subgroups between \(T\) and \(G\). One can define a structure of a graph on \(\text{Lat}(T,G)\). Also, one can define two unary operations: for \(H\in\text{Lat}(T,G)\), let \(H^{(1)}=N_G(H)\) (the normalizer of \(H\) in \(G\)) and \(H_{(1)}=T^H=\langle hTh^{-1}:h\in H\rangle\). The authors consider the situation where \(K/k\) is a finite extension of fields, \(T=\text{GL}_K(K)=K^*\) and \(G=\text{GL}_k(K)\). For an intermediate field \(L\) between \(k\) and \(K\), let \(H=\text{GL}_L(K)\). Then \(H\) is contained in a garland \(\Gamma_L\). It is proved that \(\Gamma_L=\text{Lat}(H_{(1)},H^{(1)})\).