Super Solvable Group Deficit P - Block Exists

This thesis is divided into four Chapters to obtain the sufficient and necessaryconditions of the existence of p-block of a supersolvable group with a given group.It is organized as follows:Chapter 1 is an introduction. We present the recent development of the Rep-resentation of Finite Groups, and the background of the problem discussed in thispaper.Chapter 2 is preliminaries. We give the definitions of superso1vab1e group andp-constraint group, and some basic properties of these two special groups. Then wequote the starting point of this thesis:two results in ([1], Chapter X. Lemma 6.3and Lemma 6.5 ).Chapter 3. In this chapter we first generalize the result of Ito([1],Chapter XLemma 6.5) and then obtain the sufficient and necessary conditions of the existenceof p-block of defect zero for a supersolvable group, which is a follows:Theorem 2.G is a finite supersolvable group, p is a prime and p||G|.Then G hasa p-block of defect zero if and only if O_p(G)=1.Chapter 4. In this chapter we first generalize the result of([1], Chapter X.Lemma 6.3), then on the basis of Chapter 3, we obtain the necessary and sufficienteonditlons of＊thelons exlstence ofp－blocks ofa supersolvable group tilth a9－。sz,t亿。tgYOllp,tVhlCh Is SS follotVS．Theorem 4．Sappo。,G。。a fil。。ie sop。rsol。a61。gro。p．and D d G Is a p－s。6gro。p叶G·P［．P。,…；P＝Pr,…,P。a,e an of拍。P,im,血,。z。o,s oj旧D andPI＜P,＜… ＜p。 L。tfl＝｛pr＋l,pr十。,…p。｝,Th。扣ilo w。pg g,,,q。。。al,。t· (a)．G人a。a p－block。。th d。ject g,o。p D． ／bJG人。S口n－e7eme。LJsuoh 土＊口L DDE SSg7口。。(C口(J》· C )·O。(G)二 D．...