π-Restrition, π-Induction, Kernels And Centers Of Characters

In this thesis, we obtain two main results on the base of character theory of finite groups: 7r-restriction, 7r-induction, kernels and centers of characters of finite groups.In the first part, 7r-restriction and π-constituents are defined and some of their properties are presented. Then we use it to prove a theorem about character correspondence. The main results in this part are the following theorems A and B. Theorem A offers a new technique for studying character theory and theorem B is an application of it.Theorem A Let G be a finite group and H a subgroup of G. Let X ∈ Char(G) and θ∈G Char(H). Then(2) If K is a subgroup of G with G = HK, D = H n K, thenTheorem B Let iV be a normal subgroup of a finite group G and H a subgroup of G with NH = G,N∩H = M. Suppose θ∈ Irr(N) and φ∈ Irr(M) with [θπM,φ] = 1. Assume further that IG(θ) ∩H = Ih(φ) Then π-restriction defines a bijection Irr(G|θ) —> Irr(H|φ),xIn the second part, we study the kernel and center of character. Let G be a finite group and let x be a complex character of G. Then X defines two subgroups of G: Kerx = {g ∈ G | x(g) = x(1)} and...