By Robert Hermann

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**Example text**

Then F˜ is saturated. Proof. This was first proved by Puig; cf. 3 in [P7]. 5 in [A5], where another proof appears. Here is a sketch of a different proof, based on the Roberts-Shpectorov definition of saturation. Let F˜ be a system on S˜ and let ker(α) ≤ P ≤ S be fully automized and receptive. It suffices to show that P˜ = P α is fully automized and receptive. 2 that α induces a surjective homomorphism αP : AutF (P ) → AutF˜ (P˜ ) defined by (xα)(φαP ) = xφα for φ ∈ AutF (P ) and x ∈ P . Check that AutS (P )αP = AutS˜ (P˜ ).

As the motivating example, when F = FS (G) for a finite group G, we can take Ω = G with the left and right S-actions defined by multiplication. 9] (discovered independently by Ragnarsson and Stancu [RSt, Theorem A]), a fusion system has a characteristic biset only if it is saturated. 16]: for any saturated fusion system F over a p-group S, and any subgroup P of S, the set of S-conjugacy classes of fully normalized subgroups in P F has order prime to p. 23. We are now ready to construct a transfer for fusion systems.

1. 8. Let α ∈ HomF (P, S), β ∈ HomF (P α, S), (ϕ, φ) ∈ F(α), and (Ψ, ψ) ∈ F(β). Then (a) If Q, R ≤ S, µ ∈ Φ(P, Q), and η ∈ Φ(Q, R), then µη ∈ Φ(P, R). (b) φΨ ∈ Φ(P ϕΨ, S). (c) (ϕΨ, φΨ ψ) ∈ F(αβ). Proof. 1. 9. For each P ≤ S and α ∈ homF (P, S), F(α) = ∅. Proof. 5]. Here is the idea of the proof. Choose a counter example with m = |S:P | minimal. Observe P0 = S0 as in that case (α, 1) ∈ F(α); in particular m > 1. By minimality of m, α does not extend to a proper overgroup of P in S. 8, we reduce to the case where P ∈ F f rc and α ∈ AutF (P ).