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

Abstract

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

$$ k\left( B \right) = \sum\limits_{\left( {\omega ,b_\omega } \right)} {l\left( {b_\omega } \right)} , $$

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 ) ∈ A₀(D, b _D )

• N _G (〈ω′〉, b_ω′) = N _H (〈ω′〉, b_ω′)C _G (ω′), ∀(ω′, b_ω′) ∈ (D, b _D )

If ω ₁,...,ω ₁ is a complete set of representatives of H-conjugate classes of D, then (ω ₁, b _{ω 1}), ..., (ω _l , b _{ω l} is a complete set of representatives of G-conjugate classes of B-subsections in G. In particular, we have k(B) = ∑ ^l_j=1 l(b _{ω j} ).