SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

groups A and H can be identified with subgroups of AH, and then AH is actually also
the internal semidirect product of A with H. Moreover, for every h ∈ H and a ∈ A,
the image h a of h under a via the action of A on H is just the same as a −1 ha, i.e. as
the conjugate of h by a inside the semidirect product AH. Note that AH is uniquely
determined up to isomorphism by the structure of A, H and the action of A on H.
More precisely, we have
Remark 2.1.2. Let H, ˜ H, A and à be finite groups such that A acts on H, and à acts
on ˜ H. Suppose there are group isomorphism φ : H → ˜ H and ψ : A →
à such
that (h a )φ = (hφ) (aψ) for all h ∈ H and a ∈ A. Then there is a group isomorphism
α : AH → Ã ˜ H such that α|H = φ and α|A = ψ.
As a consequence of this we get
Remark 2.1.3. Let H be a finite group, A ≤ Aut(H) and φ ∈ Aut(H). Then there is
a group isomorphism α : AH → (φ −1 Aφ)H such that α|H = φ and aα = φ −1 aφ for
all a ∈ A.
Assume from now on that G acts on a set Ω and α ∈ Ω. Then we write α G for the orbit
of α under G and |α G | for the length of this orbit. The stabilizer of α in G is denoted
by Gα. We will frequently use that |G : Gα| = |α G |. Moreover, we will apply the
following lemma:
Lemma 2.1.4 (Frattini Argument). Let N be a subgroup of G acting transitively on Ω.
Then G = GαN for every α ∈ Ω.
Using Sylow’s Theorem we get as an immediate consequence of this lemma that for
a normal subgroup N of G and P ∈ Sylp(N), G = NNG(P ). We will refer to this
property as Frattini Argument as well.
Definition 2.1.5. The action of G on Ω is said to be 2-transitive if |Ω| ≥ 2 and G acts
transitively on the set Ω (2) := {(α, β) ∈ Ω × Ω : α �= β}. Here the action of G on
Ω (2) is defined by (α, β) g = (α g , β g ) for all (α, β) ∈ Ω (2) and all g ∈ G.
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