Defect Group And Block Structure

In this paper. the author concentrates his attention on two fundamental problem in representation theory of finite groups. The first one is to determine the existence of a block with given defect group; the second one is to determine the structure of a block. At first, we give two necessary and sufficient conditions for the existence of a block with given defect group by using the graphic method and the proper- ties of Hecke algebra: then, by using several methods, we determine the existence of a block with special p梘roup as defect group( these special defect groups in- clude the submaximal subgroup of a Sylow梥ubgroup and {1}). Finally, .we prove K(B)梒onjecture and height zero conjecture for those blocks with defect group sat- isfying some fusion properties and give some characterization of the defect group of those blocks with K(B) ?L(B) = 1. The following results are the main results of this paper. Theorem 1. There is a component with D as vertex in EQ if and only if there is integer m # 0 such that the difference of the number of m ?th odd digraph and rn ?th even digraph is not 0 module p in some digraph FL with defect group D. Theorem 2. G has block of defect D if and only if there is integer rn # 0 such that the difference of the number of m ?th. odd digraph and rn ?th even digraph is not 0 module p in some digraph F with defect group D^. Theorem 3. If G contains a p梟ilpotent normal subgroup N = ED, where E = O^<sup>N. D SP(N). and Q < N. then then C has a block of defect 0 if and only if O^<sup>N has p-regular class of def€ct 0 of G. Theorem 4. IF(DLi/J(IF(Dⁱ) is ahelian. 2001 4 Theorem 5. G ha.s no p梑lock with defect group D if and only for any 14 whose defect group P^<GXGD and any element ^r 14/, the multiplicity of every element of VV in S is not prime top, where x 144. n = IGI. Theorem 6. Conditions as in (iii1^. then N has p梑lock with defect group D if and only if H = H/D has p梑lock of defect zero. Theorem 7. Conditions as in case (iv). then N has a block with defect group D if and only if there erists a p梤egular element r with defect group D such that (IA(Thi,p) = 1 if p>3 and (IAu(4)I.p) = if p = 2. Theorem 8. If G contains a p梟ilpotent normal subgroup N = ED, where F = O¹(N). D € S^N, and Q < N. then then G has a block of defect 0 if and only if O^<sup>N has p-regular class of defect 0 of G. Theorem 9. Let B be a block with defect group D, (D, i) be the Sylow b梥ubpair. If for any u, v D. they are conjugate to each other in D if and only if they are conjugate to each other in C then every virtual character of D is (G, i)-stable and if 憕 is an irreducible character of C in b with height zero, the map i?? * i is an isometry from FK(D) onto FK(G,b). Theorem 10. If k(h) ?1(b) 1 and the defect group D of b is normal in C then D is elementary ab€lian p梘roup.