Minimal characteristic bisets for fusion systems

On the basis of the Burnside ring of a fusion system

“…For each fusion system F there is a largest normal subgroup, denoted O p (F). The original proof in [5] is quite involved. In contrast the proof below, using broken chains, is actually quite elementary once you have the idea of pairing broken chains of opposite sign together.…”

“…Such formulas are important for various other applications of these bisets (see for example [12]). Using the new interpretation of these coefficients we were able to give a proof for the statement that all the stabilizers ∆(P, ϕ) appearing in Ω min must satisfy P ≥ O p (F) where O p (F) denotes the largest normal p-subgroup of F. This was originally proved in [5,Proposition 9.11], the proof we give in Proposition 7.3 uses broken chains and is much simpler.…”

“…It is shown by M. Gelvin and S. P. Reeh [5] that every characteristic biset includes a unique minimal characteristic biset, denoted by Ω min . The minimal biset can be described as the basis element α ∆(S,id) of the fusion system A(F × F), where for a morphism ϕ : Q → S in F, the subgroup ∆(P, ϕ) denotes the diagonal subgroup {(ϕ(s), s) | s ∈ P } in S × S. Now Theorem 1.2 can be used to give formulas for the coefficients c ∆(P,ϕ) (Ω min ).…”

“…In [9] it is shown that the exists a characteristic biset for F if and only if F is saturated, and it is shown how to reconstruct F given any F-characteristic biset. In [5] two of the authors of this paper give a parametrization of all the characteristic bisets for a given saturated fusion system F. In particular it is shown that there is a unique minimal Fcharacteristic biset Λ F , and every other F-characteristic biset contains at least one copy of Λ F . Theorem 7.2 ([5, Theorem 5.3 and Corollary 5.4]).…”