New properties of the \(\pi\)-special characters (Q1355654)

In character theory it is important to know as much as possible about how one may construct new irreducible characters from old ones. Induction of characters usually yields interesting aspects concerning such a construction. Not mentioning definitions here, we note that the aim of the paper is to prove Theorem A: Suppose \(\mathcal G\) is solvable of odd order. Let \({\mathcal H}\leq{\mathcal G}\) and suppose \(\alpha\in\text{Irr}({\mathcal H})\) is a \(\pi\)-special character for which the induced character \(\alpha^{\mathcal G}\) is irreducible. Suppose \(\beta\in\text{Irr}({\mathcal H})\) is \(\pi'\)-special. Then \((\alpha\beta)^{\mathcal G}\in\text{Irr}({\mathcal G})\). In order to prove Theorem A, much work is done here interesting on its own account.