On the control of fusion in the local category for the p-block with a minimal nonabelian defect group

作     者：YANG Sheng GAO Sheng 

作者机构：School of Mathematical Sciences Peking University Beijing 100871 China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2011年第54卷第2期

页      面：325-340页

核心收录：

学科分类：083002[工学-环境工程] 0830[工学-环境科学与工程（可授工学、理学、农学学位）] 07[理学] 08[工学] 09[农学] 0903[农学-农业资源与环境] 0712[理学-科学技术史(分学科，可授理学、工学、农学、医学学位)] 0713[理学-生态学] 

主　　题：defect group subpair control of fusion Brauer's net 

摘      要：If B is a p-block of a finite group G with a minimal nonabelian defect group D (p is an odd prime number) and (D, b D ) is a Sylow B-subpair of G, then N G (D, b D ) controls B-fusion of G in most cases. This result is of great importance, because we can use it to obtain a complete set of representatives of G-conjugate classes of B-subsections and to calculate the number of ordinary irreducible characters in B. This result is key to the calculation of the structure invariants of the block with a minimal nonablian defect group. On the other hand, we improve Brauer s famous formula k(B) =Σ (ω,b ω ) l(b ω ),where (ω, b ω ) ∈ [(G : sp(B))]. Let p be any prime number, B be a p-block of a finite group G and (D, b D ) be a Sylow B-subpair of G. H is a subgroup of N G (D, b D ) satisfying N G (R, b R ) = N H (R, b R )C G (R), (R, b R ) ∈ A 0 (D, b D ), N G ( w , b w ) = N H ( w , b w )C G (w ), (w , b w ) ∈ (D, b D ). If w 1 , . . . , w l is a complete set of representatives of H-conjugate classes of D, then (w 1 , b w 1 ), . . . , (w l , b w l ) is a complete set of representatives of G-conjugate classes of B-subsections in G. In particular, we have k(B) =Σ l j=1 l(b w j ).