Adams operations on character restrictions (Q804693)

Let G be a finite group of order n, let p be a prime, and let P be a Sylow p-subgroup of G. We denote by M the set of all linear combinations of irreducible characters of G with coefficients in \({\mathbb{Z}}_{(p)}:=\{a/b:\) a,b\(\in {\mathbb{Z}}\), \(p\nmid b\}\), and by N the set of restrictions to P of elements in M. The multiplicative group \(\Gamma:={\mathbb{Z}}/n{\mathbb{Z}}^{\times}\) acts on N in such a way that \(\chi^{k+n{\mathbb{Z}}}(g)=\chi (g^ k)\) for \(\chi\in N\), \(g\in P\), \(k+n{\mathbb{Z}}\in \Gamma\). The main result of the paper asserts that in this way N becomes a permutation module over \({\mathbb{Z}}_{(p)}\Gamma\). This gives a positive answer to a question raised by \textit{K. Knapp} [Comment. Math. Helv. 63, 414-449 (1988; Zbl 0677.55003)].