On linear groups with a cyclic or T.I. Sylow subgroup
Let G be a finite group with cyclic Sylow p-subgroup P. Let \(\chi\) be a faithful character of G. The author shows that if \(| P| >p\) and G is not p-solvable then \(\chi\) (1)\(\geq | P| -2\). Suppose that P is cyclic. Then if G is still not p-solvable \(\chi\) (1)\(\geq | P| -1\) and G/Z(G) is more or less alternating or special linear or projective special linear. If \(\chi\) is faithful and G is p-solvable then \(\chi\) (1) is almost as big as \([P:O_ p(G)]\) depending on the largest of the defect groups of the irreducible constituents of \(\chi\). If instead of cyclic P is T.I. and \(\chi (1)<| P| -1\) the \(O^{p'}(G)\) mod its center is usually in another list the author gives.
scientific article; zbMATH DE number 4040163
scientific article