A Brauer-Wielandt formula (with an application to character tables) (Q2816999)

The Brauer-Wielandt formula referred to in the title appears in [\textit{H. Wielandt}, Math. Z. 73, 146--158 (1960; Zbl 0093.02302)]. The paper proves a different but related formula. Namely, let \(P\) be any finite \(p\)-group (\(p\) any prime) acting on a finite group \(G\) of order prime to \(p\). Then NEWLINE\[NEWLINE \left|\mathbf{C}_G(P)\right| = \left(\prod_{x \in P}\frac{\left|\mathbf{C}_G(x)\right|}{\left|\mathbf{C}_G\left(x^p\right)\right|^{1/p}}\right)^{\frac{p}{(p -1)|P|}}. NEWLINE\]NEWLINE They use the new formula to deduce that if \(p\) is any prime, \(G\) is a finite \(p\)-solvable group, and \(P\) is a Sylow \(p\)-subgroup of \(G\) which is abelian or of exponent \(p\), then the character table of \(G\) determines \(|\mathbf{N}_G(P)|\). In a similar manner, they conclude that if \(G\) is a finite \(p\)-solvable group with arbitrary Sylow \(p\)-subgroup \(P\), the character table of \(G\) plus the \(p\)-power map determine \(|\mathbf{N}_G(P)|\).