A subgroup H of a finite group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H n X = 1. In this easel K is called a complement of H in G. In this note some results on complem...A subgroup H of a finite group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H n X = 1. In this easel K is called a complement of H in G. In this note some results on complemented subgroups of finite groups are obtained.展开更多
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 i...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 ).展开更多
基金Project supported by the National Natural Science Foundation of China (No.19671073) and theYouth Science Foundation of Shanxi
文摘A subgroup H of a finite group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H n X = 1. In this easel K is called a complement of H in G. In this note some results on complemented subgroups of finite groups are obtained.
文摘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 ).
