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

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) = Sigma((omega,b omega)) l(b(omega)), where (omega, b(omega) ) is an element of [(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), for all(R, b(R)) is an element of A(0)(D, b(D)), N-G (<omega'>, b(omega')) = N-H (<omega'>, b(omega'))C-G (omega'), for all(omega', b(omega')) is an element of (D, b(D)). If omega(1), ... ,omega(1) is a complete set of representatives of H-conjugate classes of D, then (omega(1), b(omega 1)), ... , (omega(l) , b(omega l)) is a complete set of representatives of G-conjugate classes of B-subsections in G. In particular, we have k(B) = Sigma(l)(j=1) l(b(omega j)).Mathematics, AppliedMathematicsSCI(E)2ARTICLE2325-3405