Characters of groups with quotients of odd order (Q1059708)

The following theorem is proved: Let G be a group of order g. Let \(H\triangleleft G\), \([G:H]=m\), m being odd. Let \(\tau \in Gal({\mathbb{Q}}_ g/{\mathbb{Q}})\) be such that \(q^{\tau}=\bar q\), for all \(q\in {\mathbb{Q}}_ m\). Let \(\psi\in Irr(H)\) be fixed by \(\tau\). There exists a unique irreducible constituent of \(\psi^ G\), \(\chi\), such that \(\chi\) is fixed by \(\tau\). Further, \({\mathbb{Q}}(\chi)\subseteq {\mathbb{Q}}(\psi^ G)\). There are four corollaries. A remarkable one is this. Corollary. If every simple group of even order has an irreducible rational character of odd degree, other than the trivial, then the same is true for all groups of even order.