Stabilizer Limits Of π-partial Characters

In this thesis,we generalize some important results about stabilizer limits of complex characters to π-partial characters due to Isaacs.We discuss the correspondence of characters between Iπ-triples and their Clifford reductions.In particular,we establish a natural correspondence between the stabilizer limits of an Iπ-triple T and a cover subtriple of T.Our first main result gives a correspondence between characters of an Iπ-triple and its multi-Clifford reductions.Theorem A.Suppose that T=(G,N,ψ)is an Iπ-triple.If T’=(G’,N’,ψ’)is a multiClifford reduction of T,and H is a subgroup of G containing N,then H’=G’∩H is a subgroup of G’ containing N’ and induction defines a bijection Iπ(H’|ψ’)→Iπ(H|ψ).The second main result establishes a correspondence between the set of all subtriples of the Iπ-triple and all subtriples of a cover subtriple of the Iπ-triple.Theorem B.Let T=(G,N,ψ)be an Iπ-triple,and let T=(G,N,ψ)be a subtriple of T covering T.Then there is a correspondence between the set of all subtriples of T and T,lying above Z(T)and Z(T).Here two such T’=(G’,N’,ψ’)and T’=(G’,N’,ψ’)correspond if and only if the subgroups G’ and N’ containing Z(T)correspond respectively to the subgroups G’,N’ containing Z(T),also the irreducible π-partial character ψ’∈Iπ(N’|ζ)corresponds to ψ’∈Iπ(N’|ζ).As applications,we obtain the following three main theorems.Theorem C.Let T be an Iπ-triple,and let T be a subtriple of T covering T.If a subtriple T’ of T lying above Z(T)corresponds to a subtriple T’ of T lying above Z(T),then any Clifford reduction of T’ is a subtriple of T above Z(T),and corresponds to some Clifford reduction of T’.Similarly,any Clifford reduction of T’ is also a subtriple of T above Z(T)and corresponds to some Clifford reduction of T’.This result shows that there is a natural correspondence between the Clifford reductions of an Iπ-triple T and these of a cover subtriple of T.Theorem D.Let T be an Iπ-triple,and let T be a subtriple of T covering T.Then any multiClifford reduction of T above the center corresponds to some multi-Clifford reduction of T.Similarly,any multi-Clifford reduction of T above the center corresponds to some multiClifford reduction of T.This result establishes a correspondence between the multi-Clifford reductions of a Iπtriple and the multi-Clifford reductions of its cover subtriples.Theorem E.Let T be an Iπ-triple,and let T be a subtriple of T covering T.Then the stabilizer limit of T is above the center of T and corresponds to the stabilizer limit of T.If the stabilizer limit T’ of T corresponds in this way to the stabilizer limit T’ of T,then the sub triple T’ covers T’.This result describes a natural correspondence between the stabilizer limits of the Iπtriples and those of its cover subtriples.The main results of this thesis provide a new proof technique for studying the relationship between π-partial characters of π-separable groups and π-partial characters of subgroups.In particular,when π is the coset of a prime p,the corresponding Brauer character standard version can be obtained.The above theorems can be used to study complex characters or Brauer characters.