\(p\)-Steinberg characters of finite simple groups
Let \(p\) be a prime number and \(G\) be a finite group such that \(p\mid|G|\). A \(p\)-Steinberg character of \(G\) is an irreducible complex character \(\chi\) of \(G\) such that \(\chi(x)=\pm|C_G(x)|_p\) for every \(p'\)-element \(x\) of \(G\), where \(n_p\) denotes the \(p\)-part of the natural number \(n\). The notion of a \(p\)-Steinberg character was introduced by \textit{W. Feit} [in Linear algebraic groups and their representations, Los Angeles 1992, Contemp. Math. 153, 1-9 (1993; Zbl 0823.20013)] in which the following conjecture is made: Let \(p\) be a prime number and \(G\) a finite simple group such that \(p\mid|G|\). If \(G\) has a \(p\)-Steinberg character, then does it follow that \(G\) is isomorphic to a simple group of Lie type defined in characteristic \(p\)? Feit suggests using the classification of finite simple groups the above question may be answered affirmatively. Using the character tables of the sporadic simple groups [in \textit{J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker} and \textit{R. A. Wilson}, Atlas of finite groups (Clarendon, Oxford, 1985; Zbl 0568.20001)] it is easy to see that none of the 26 sporadic simple groups has a \(p\)-Steinberg character. For the alternating and the projective special linear groups, the reviewer proved W. Feit's conjecture [in J. Algebra 181, No. 1, 196-206 (1996; Zbl 0851.20008)]. In the paper under review the author proves the above mentioned conjecture of Feit for the rest of the finite simple groups.
scientific article; zbMATH DE number 1014045
scientific article