π-Induction Of Characters

In the character theory of finite groups,it is a fundamental and important question to study the irreducible induction of characters from subgroups.In 1997 Navarro proved three theorems for irreducible induction of π-special characters of subgroups in odd-order groups,which have important applications in Isaacs’ π-theory.In this thesis,we will remove the condition of odd-order groups,and use the π-induction of characters in π-separable groups to replace the usual induction,and prove three similar results regarding the irreducible π-induction of the special characters.Our results will have more applications.The main results of this thesis are as follows:Theorem A.Suppose that G is a π-separable group,where 2 /∈ π.Let H ≤ G andα ∈ Xπ（H）.Suppose that απGis irreducible.Then for any β ∈ Xπ′（H）,（αβ）^πGis irreducible.The following Theorem is our principal tool for proving the Theorem A.Theorem B.Suppose that G is a π-separable group,where 2 /∈ π.Let H ≤ G andα ∈ Xπ（H）such that απG∈ Irr（G）.If J ≤ G and |H : H ∩ J| is a π′-number,then（αH∩J）πJ∈ Irr（J）.Finally,in order to discuss whether the map π-induction in Theorem A is injection,we introduce Dπ-nucleus of Dπ-character（for definitions and properties,see the preliminaries in this paper）,then we can obtain the following Theorem C.Theorem C.Suppose that G is a π-separable group,where 2 /∈ π.Let（W,γ）be a nucleus of χ ∈ Dπ（G）and let δ₁,δ₂∈ Xπ′（W）.If（γδ₁）^πG=（γδ₂）^πG,then δ₁= δ₂.