On a conjecture concerning character degrees of some \(p\)-groups
It has been conjectured that the character degrees of \(P\in\text{Syl}_p(A_n(q))\), \(q\) a \(p\)-power, are \(q\)-powers. This has recently been established by \textit{I. M. Isaacs} [J. Algebra 177, No. 3, 708-730 (1995; Zbl 0839.20010)]. We extend this result to the \(p\)-Sylow subgroups of \(C_n(q)\), \(D_n(q)\) and \({^2A}_{2n-1}(q^2)\), \(p\) an odd prime. This result was proved by \textit{D. Kazhdan} [Isr. J. Math. 28, 272-286 (1977; Zbl 0391.22006)] under the restriction \(n<p\), which allows him to define the exponential map on the Lie algebra \(L(P)\) associated to \(P\). On the other hand, Isaacs shows that the character degrees of \(P\) are \(q\)-powers, whenever \(P\) is an algebra group over \(\mathbb{F}_q\). Unfortunately, it is not clear whether this applies to our cases. To solve this problem, we define a map sharing some of the properties of the exponential map, with the advantage of being defined whenever \(p\) is odd, and prove that some suitable sections of \(P\) are strong subgroups.
scientific article; zbMATH DE number 937024