?-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 ?-special characters of subgroups in odd-order groups,which have important applications in Isaacs' ?-theory.In this thesis,we will remove the condition of odd-order groups,and use the ?-induction of characters in ?-separable groups to replace the usual induction,and prove three similar results regarding the irreducible ?-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 ?-separable group,where 2 /? ?.Let H ? G and? ? X??H?.Suppose that ??Gis irreducible.Then for any ? ? X???H?,????^?Gis irreducible.The following Theorem is our principal tool for proving the Theorem A.Theorem B.Suppose that G is a ?-separable group,where 2 /? ?.Let H ? G and? ? X??H?such that ??G? Irr?G?.If J ? G and |H : H ? J| is a ??-number,then??H?J??J? Irr?J?.Finally,in order to discuss whether the map ?-induction in Theorem A is injection,we introduce D?-nucleus of D?-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 ?-separable group,where 2 /? ?.Let?W,??be a nucleus of ? ? D??G?and let ?₁,?₂? X???W?.If???₁?^?G=???₂?^?G,then ?₁= ?₂.