The permutation index of p-defect zero characters (Q1099246)

Let G be a finite group and let \(\chi\in Irr(G)\). Let \[ sp(\chi)=\sum_{\sigma \in Gal({\mathbb{Q}}(\chi)/{\mathbb{Q}})}\chi^{\sigma}. \] It is a theorem of E. Artin that \(| G| sp(\chi)\) is a \({\mathbb{Z}}\)-linear combination of permutation characters. Denote by n(\(\chi)\) the least positive integer so that n(\(\chi)\)sp(\(\chi)\) is such a combination. Now, if p is a prime not dividing \(| G| \chi (1)^{-1}\) (\(\chi\) has p-defect zero) it is a conjecture that \(p\nmid n(\chi)\). Four of the six main theorems of the paper are as follows. A. If G is supersolvable and \(\chi\) has p-defect zero, then \(p\nmid n(\chi).\) B. If G is solvable and \(\chi\) (1) is the p-part of \(| G|\), then \(p\nmid n(\chi).\) D. If \(| G|\) is odd and every q-Sylow subgroup of G is abelian for each \(q\neq p\), and \(\chi\) has p-defect zero then \(p\nmid n(\chi).\) E. If M is a normal abelian subgroup of G and G/M is supersolvable, and \(\chi\) has p-defect zero, then \(p\nmid n(\chi)\).