Linear characters of Sylow subgroups.
In this paper, the author was motivated by considering the McKay conjecture for finite groups with a self-normalizing Sylow \(p\)-subgroup. If \(p\geq 5\), the author clarifies the situation. Here are a few of the results in the paper: Theorem A. If \(\lambda,\mu\) are linear characters of a finite \(p\)-group \(P\) with \(p\) an odd prime and \(\lambda=\mu\) up to induced characters of proper subgroups, then \(\lambda=\mu\). Corollary B. Let \(P\) be a Sylow \(p\)-subgroup of \(G\) with \(p\) odd. If \(P\) is self-normalizing, \(\lambda,\mu\) are linear characters of \(P\) and \(\lambda^G=\mu^G\), then \(\lambda=\mu\). The author shows that certain nice class functions are characters (Theorem~D and Corollary~E). Theorem F. Suppose that \(G\) is \(p\)-solvable and \(P\) is a self-normalizing Sylow \(p\)-subgroup. (a) If \(\lambda\) is a linear character of \(P\), then \(\lambda^G\) is the sum of an irreducible character of degree prime to \(p\) and irreducible characters of degree divisible by \(p\). (b) Let \(\chi\) be an irreducible character of \(G\) of degree prime to \(p\). Then \(\chi_P\) is the sum of a linear character and nonlinear irreducible characters of \(P\). Furthermore the map \(\lambda\to\chi\) is a bijection of the linear characters of \(P\) with the set of irreducible characters of \(G\) of degree prime to \(p\). Theorem F is false without assuming solvability (for \(p=2\)). Moreover, by a result of the author, the reviewer and \textit{G. Malle}, Self-normalizing Sylow subgroups [Proc. Am. Math. Soc. 132, No. 4, 973-979 (2004; Zbl 1049.20010)], \(P\) self-normalizing implies that \(G\) is solvable (using the classification of finite simple groups).
scientific article; zbMATH DE number 2013030