Minimal modular character degrees for groups with a cyclic Sylow subgroup (Q910495)

Let F be a field of prime characteristic p, and let G be a finite linear group over F with a cyclic Sylow p-subgroup P such that \(| P| -1\) exceeds the degree of G. The main result of the paper under review shows that \(O^{p'}(G)O_{p'}(G)/O_{p'}(G)\) is isomorphic to either P, PSL(2,p), \(A_ 7\) \((p=7)\), \(J_ 1\) \((p=11)\), \(A_ 6\) \((p=5)\), \(A_ 7\) \((p=5)\), \(A_ p\), \(M_{11}\) \((p=11)\), \(M_{23}\) \((p=23)\), or PSL(n,q) where n is a prime not dividing q-1 \(and| P| =(q^ n-1)/(q-1)\). (The actual statement given in the paper is more precise.) This theorem extends earlier results by Brauer, Tuan, Feit, Ferguson and the author. Its proof makes use of the classification of finite simple groups. As an application the author classifies finite groups with a cyclic Sylow p- subgroup such that the projective cover of the trivial module in characteristic p has dimension p.