Character Correspondences And Nilpotent Fully-Ramified Sections Under Group Actions

In 1973, Isaacs established character theory of abelian fully-ramified basic configuration (G,K,L,ε,φ), especially constructed a magic character and used it to describe the character correspendence . In 1997, Lewis promoted it to coprime nilpotent fully-ramified basic configuration, constructed Lewis magic character and character correspondence related to it. In this paper, we further generalize the work of Lewis.we mainly discuss the construction of magic character and character correspondence problem under the action of operator groups. Especially,we prove that both magic charac-terand Lewis correspondence are admmit with operator groups.The following are main results of this thesis:Theorem 2.2 Let (G, K, L) be a coprime or a controlled normal triple with complement H. K/L is solvable, S act on G and K, L, H are all S'-invariant, (|S|, |K : L|)—1, φ G Irrs(L) is H-invariant, then there exists a S'-invariant and chief H-complete chain C in K starting with (L,φ). Furthermore , if ε∈Irr(K|φ) is H-invariant and S-invariant, then we may choose a S-invariant and chief H-complete chain C so that its cover is e.Corollary 2.3 Let (G, K, L) be a normal triple with complement H, S act on G and K, L, H are all S-invariant, φ∈Irrs(L) is H-invariant, Let C be a S'-invariant H-complete chain in K starting with (L,φ), then:(1) C is a S-invariant H-complete chain in Ko starting with (L,φ );(2) C¹ is a S-invariant H Li-complete chain in Ko starting with (L₁, φ₁);(3) if K/L is nilpotent, then a S'-invariant and chief H-complete chain is chief.Theorem 2.4 Let (G, K, L) be a normal triple with complement H such that |G : K| or |K : L| is odd, S act on G and K,L,H are all S-invariant, φ∈ Irrs(L) is H-invariant, Let C be a S-invariant H-complete chain in K starting with (L, φ) and having cover ε, Then we construct a bijection, Δ_c^G : Irr(H|φ) → Irr(G|ε) such that(1) Δ_c^G is degree proportional, that is x(1)/θ(1) = ε(1)/φ(1) whenever θ ∈ Irr(H|φ) and X = Δ_c^G(θ)(2) Δ_c^G and S is abelian, That is for s ∈ S and θ € Irr(H|φ ), we have (Δ_c^G(θ)⁾s = Δ_c^G(θ^s). Furthermore, if θ∈Irr(H|φ) is S-invariant and x = Δ_c^G(θ), then x is S-invariant.(3) if \G : L\ is odd, then θ is a constituent ofTheorem 2.5 Let (G,K,L,ï¿¡,tp) be a nilpotent fully-ramified coprime basic configuration with stabilizing complement H. S act on G and K,L,H are all S-invariant, (\S\, \K : L|)=l, then we can define a S-invariant V ï¿¡ Char(if) such that(l)the equation xh — ^ for x ï¿¡ ln{G\e) and 6 € lxr{H\(p) defines a one-to-one correspondence between these sets of characters;(2)if \G : L\ is odd, then 6 is the unique constituent of xh having odd multiplicity.