Some Brauer Versions Of Dolfi Theorems

In 2002, Dolfi extended the Clifford theorem of complex character theory, and obtained three main results by replacing the normal subgroup conditions with some arithmetical conditions. In this paper, we will further expand the Dolfi theorems into Brauer character theory, and get some Brauer versions of Dolfi theorems, which also extend the Clifford theorem in Brauer character theory correspondently.The following is the first main result of this thesis:Theorem 2.4 Let G be a p-solvable group, x ∈ IBr_p(G) an irreducible Brauer character, N,K ≤L≤G with N(?)L, K(?)G. Assume that (x(1), |G : L||N : N∩K|) = 1. Then(1) all the irreducible constituents of x_N are L-conjugate;(2) if φ ∈ IBr_p(N) is an irreducible constituent of x_N, then φ_N∩K ∈ IBr_p(N∩K).This theorem is equal to the Brauer version of Dolfi theorem A, another equivalent form is the following, which can be treated as a direct extension of Clifford theorem in Brauer theory:Theorem 2.5 Let G be p-solvable with a subgroup H. Let x ∈ IBr_p(G) be an irreducible Brauer character of G. If (x(1),|G : N_G(H)||H : Core_G(H)|) = 1, thenwhere e is a nonnegative integer, φ_i ∈ Irr(H) are the distinct conjugates of θ in N_G(H).The second main result of this thesis is about π-degrees of Brauer characters, it is equivalent to the Brauer version of Dolfi theorem B:Theorem 2.6 Let G be a p-solvable group, N (?)L ≤ G, x ∈ IBr_p(G), π = π(x(1)). If |O^π(G)L : L||N : N ∩ O_π(G)| is a π'-number, then all the irreducible constituents of x_N have π-degrees.In the third main result, we will study when the restriction to a subgroup of an irreducible π-partial character is also irreducible, it can be treated as the π-form of Dolfi theorem C:...