On the connected components of the spectrum of the extended character ring of a finite group. (Q2370239)

Let \(G\) be a finite group with character ring \(R(G)\), let \(S\) be a subring of \(\mathbb{C}\) containing a primitive \(|G|\)-th root of unity, and let \(\pi\) be the set of prime numbers \(p\) such that \(p^{-1}\notin S\). Making use of arguments in Serre's book on representation theory, the authors prove the following variant of Brauer's Induction Theorem: \(S\otimes_\mathbb{Z} R(G)=\sum_H\text{Ind}^G_H(R(H))\) where \(H\) ranges over the set of \(p\)-elementary subgroups of \(G\), for \(p\in \pi\). (When \(\pi=\emptyset\), the sum ranges over the cyclic subgroups of \(G\).) As a consequence, the authors show that the connected components of the prime spectrum of \(S\otimes_\mathbb{Z} R(G)\) (or, equivalently, the primitive idempotents of \(S\otimes_\mathbb{Z} R(G)\)) are in bijection with the \(\pi\)-regular conjugacy classes of \(G\). This also leads to a proof of a theorem of Frobenius: For every divisor \(n\) of \(|G|\), the number of elements \(x\in G\) such that \(x^n=1\) is a multiple of \(n\).