A note on degrees of irreducible \(\pi\)-Brauer characters.
Let \(G\) be a finite \(\pi\)-separable group where \(\pi\) is a set of primes. Isaacs and others have developed a theory of Brauer characters with respect to \(\pi\). The authors prove: (1) \(G\) has a normal Hall \(\pi\)-subgroup if and only if every irreducible \(\pi\)-Brauer character of \(G\) has \(\pi'\)-degree. (2) If \(G\) has an Abelian \(\pi\)-complement then every irreducible \(\pi\)-Brauer character of \(G\) has \(\pi\)-degree. (3) \(|G|_{\pi'}\leq\sum_{\varphi\in\text{IBr}_\pi(G)}\varphi(1)^2\), with equality if and only if \(G\) has a normal Hall \(\pi\)-subgroup. These results are analogs of known results for the classical case where \(\pi\) consists of a single prime \(p\).
scientific article; zbMATH DE number 5500574