A Note on the Prime Spectrums of Character Rings of Finite Groups

摘要

Let G be a finite group with order g and S be a subring of the algebraic number field which contains the integral extension over Z generated by a g-th primitive root omega of unity, and R(G) be the character ring of G, The prime spectrum of the commutative ring S(X)_z-R(G) is denoted by Spec(S (X)_z-R(G)) and set pi = {p|p is a rational prime number such that p~(-1) S}. We prove that when G is a pi-group, a pi'-group, or a finite Abelian group, the number of the connected components of Spec(S(x)_z R(G)) coincides with the number of the pi-regular classes in G.