Irreducible induced characters of solvable groups
The paper considers relationships between certain normal subgroups of a finite group G and normal subgroups of a subgroup \(H\subseteq G\) in the presence of a character \(\theta\) of H such that \(\theta^ G\) is irreducible. The main theorem is: Let \(\pi\) be a set of primes and let G be a finite \(\pi\)-solvable group. Let H be a subgroup of G and suppose that \(\theta\in Irr(H)\) with \(\theta^ G\in Irr(G)\). Then \[ O_{\pi '}(G)\subseteq \ker \theta^ G\quad \Leftrightarrow \quad O_{\pi '}(H)\subseteq \ker \theta. \] Furthermore, if 2\(\not\in \pi\), then \[ O_{\pi '}(G)\subseteq \ker \theta^ G\quad \Rightarrow \quad O_{\pi '}(H)=O_{\pi '}(G). \] A corollary to this theorem says that \(\theta\) is in the principal \(\pi\)-block of H if and only if \(\theta^ G\) is in the principal \(\pi\)-block of G, where \(\pi\)-blocks are defined as in the author's thesis. A straightforward generalization of the proof of this theorem is also given which deals with an arbitrary Fitting class: Theorem. Let G be a finite solvable group and let \({\mathcal F}\) be any Fitting class. We will write \(O_{{\mathcal F}}(G)\) to denote the largest normal subgroup of G contained in \({\mathcal F}\). Suppose that \(H\subseteq G\) and \(\theta\in Irr(H)\) with \(\theta^ G\) an irreducible character of odd degree. Then \[ O_{{\mathcal F}}(G)\subseteq \ker \theta^ G\quad \Rightarrow \quad O_{{\mathcal F}}(H)=O_{{\mathcal F}}(G). \]
scientific article; zbMATH DE number 3843983