On the connected components of the spectrum of the extended character ring of a finite group

Abstract

Let R(G) be the character ring of a finite group G. For any subring S of the complex field, let π be the set of such rational primes whose inverse do not lie in S, and consider the prime spectrum of the coefficient extended character ring S ⊗ R(G). By means of a generalized version of Brauer's induction theorem, we show that the number of the connected components of the prime spectrum of S ⊗ R(G) equals the number of the π -regular conjugacy classes of G.

On the connected components of the spectrums of the character rings of a finite group

“…Just like many related works, e.g. [1,3], such a construction of characters plays an important role in the proof of Theorem 1.1, which is completed in Section 3. However, in Section 3 we'll first prove Theorem 3.3 which is in fact a restatement of Corollary 1.2 with the idempotents described above; and then we'll deduce Theorem 1.1 from it.…”

“…(i) (See [3].) The number of connected components of Spec(Z π [ω h ] ⊗ R(G)) is equal to the number of π -regular classes of G.…”

“…Ref. [3] shows that the number of connected components of Spec(Z π [ω h ] ⊗ R(G)) is equal to the number of π -regular classes of G. For the characters over the field K we have to consider the following Galois group.…”