The depth of subgroups of \(\mathrm{PSL}(2,q)\). (Q420683)

Let \(H\) be a subgroup of a finite group \(G\), and let \(\text{Irr}(G)=\{\chi_1,\dots,\chi_s\}\), \(\text{Irr}(H)=\{\psi_1,\dots,\psi_r\}\). Then the \(r\times s\)-matrix \(M=(m_{ij})\) with \(m_{ij}:=(\psi_i^G,\chi_j)\) for \(i=1,\dots,r\), \(j=1,\dots,s\) is called the inclusion matrix of \(H\) in \(G\). Let \(M^{(1)}:=M\) and \(M^{(2k)}:=M^{(2k-1)}M^T\), \(M^{(2k+1)}:=M^{(2k)}M\) for \(k\geq 1\). The depth \(d(H,G)\) of \(H\) in \(G\) is defined as the smallest integer \(n\) such that \(M^{(n+1)}\leq cM^{(n-1)}\) for some positive integer \(c\). Here \(A\leq B\) for two integer matrices \(A=(a_{ij})\), \(B=(b_{ij})\) of the same size means that \(a_{ij}\leq b_{ij}\) for all \(i,j\). This notion of depth has its origin in the theory of von Neumann algebras. It is known that \(d(H,G)\leq 2\) if and only if \(H\) is a normal subgroup of \(G\).NEWLINENEWLINE In the paper under review, the author computes \(d(H,G)\) for all subgroups \(H\) of \(G=\text{PSL}(2,q)\). It turns out that \(d(H,G)\leq 5\) in all cases, and \(d(H,G)=3\) in most cases.