On the dimensions of PIM's. (Q2865897)

Let \(G\) be a finite group with Sylow \(p\)-subgroup \(P\), for a prime \(p\), and let \(\text{IBr}_p(G)\) be the set of irreducible \(p\)-Brauer characters of \(G\). For \(\varphi\in\text{IBr}_p(G)\), let \(\Phi_\varphi\) denote the corresponding principal indecomposable character, and set \(c_\varphi:=\Phi_\varphi(1)/|P|\). The authors show that \(G\) is \(p\)-solvable if \(c_\varphi=\varphi(1)_p\), for all \(\varphi\in\text{IBr}_p(G)\). This is the converse of a result by P. Fong. (Here \(n_p\) denotes the \(p\)-part of a positive integer \(n\).) They also prove that \(G\) is \(p\)-solvable if \(c_\varphi=\varphi^{\text{Fix}(P)}(1)\) for all \(\varphi\in\text{IBr}_p(G)\); here \(\varphi^{\text{Fix}(P)}(1)\) is the dimension of the vector space of \(P\)-fixed points on the module affording \(\varphi\). Several related results and conjectures appear in the paper under review.