Remarks on coherence and the Reynolds isometry (Q762604)

Let G be a finite group, p a prime and H a subgroup of G. Suppose that H contains a Sylow p-subgroup P of G such that, for every non-identity subgroup Q of P, \(N_ G(Q)=N_ H(Q)\cdot O_{p'}(C_ G(Q))\). Then H controls strong p-fusion and \textit{W. F. Reynolds} [Math. Z. 107, 264-270 (1968; Zbl 0174.054)] has proved the existence of an isometry \(\tau\) : \(V_ 0^{(p)}(H)\to V_ 0^{(p)}(G)\), where \(V_ 0^{(p)}(H)\) is the space of complex-valued class-functions on H which vanish on p- regular classes and are linear combinations of irreducible characters in the principal p-block \(B_ 0^{(p)}\) of H, onto the corresponding space for G. The mapping \(\tau\) sends generalized characters to generalized characters. The relationship between \(B_ 0^{(p)}(H)\) and \(B_ 0^{(p)}(G)\) is further analyzed here. The author shows that if \(\chi\),\(\mu\) are irreducible characters in \(B_ 0^{(p)}(G)\), \(B_ 0^{(p)}(H)\), respectively, and the values of \(\chi\),\(\mu\) on certain elements of H satisfy congruence relations (modulo certain powers of p), then their values must be equal. (The details are a little too technical to include here.) As an application, the author shows how to get fusion results for odd primes p, similar to the Glauberman \(Z^*\)-theorem for \(p=2\).