Fusion systems

Abstract

Abstract. This is a survey article on the theory of fusion systems, a relatively new area of mathematics with connections to local finite group theory, algebraic topology, and modular representation theory. We first describe the general theory and then look separately at these connections.

Fusion systems over a Sylow p-subgroup of $${\text {G}}_2(p)$$ G 2 ( p )

“…To see that this group is unique, we show that θ is uniquely determined as a subgroup of Z(Γ), θ and this is where we hypothesis (2). In the case that 3 does not divide p −1, we have that Z(Γ), θ acts faithfully on Z 2 because the elements of Z(Γ) scale V by some ω ∈ F 5 and then Z by ω 2 (so the determinant 1 elements in Z(Γ) have order dividing 3).…”

The Local Structure Theorem, the non‐characteristic 2 case

“…Again we have U Q K = y × K and so U Q ∩ K is an elementary abelian subgroup of K normalized by L K . We deduce Since (2). Now y M has size 1, 7 or 8.…”

“…Since Inn(X) acts transitively on the involutions in X/Z(X), N Aut(X) (Y )Inn(X) = Aut(X). As C Inn(X) (Y ) = T , and |Y | = 2 3 , the subgroup structure of SL 3 (2)…”

“…Suppose that K/Z(K) ∼ = Ru. Then by [6, Table 5.3r] there is a 2-local subgroup J of K containing S y with J/Z(K) ∼ 2 3+8 · SL 3 (2) and J is normalized by O 2 (M ). Hence Suppose that K ∼ = 2 F 4 (2) .…”

“…Assume that Z(K) = 1. Then, by [6, (2). Further the preimage of Z(S y /Z (K)) is isomorphic to Z 4 × Z 2 .…”