SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

Therefore, it will be consistent to denote both by h τ . Similarly we identify H and σ
with there images in 〈σ〉 ⋉ H.
If (*) holds then σ α = τch for some h ∈ H and Z(H) = 1. It follows o(τch) =
o(σ) = 2. This gives us ˆ h = ˆ(τch) 2
h
= ˆ h τhτh for every ˆ h ∈ H. Thus, h τ h = τhτh ∈
Z(H) = 1. Therefore, h τ = h −1 . So we may assume from now on that (**) holds. By
2.9.2, it is sufficient to show the following.
(+) There exists a group isomorphism α ∗ : 〈σ〉H → 〈τ〉H such that α ∗ |H = α|H.
By (**), we have o(τh) = 2 inside of 〈τ〉H. This yields 〈τ〉H = 〈τh〉 ⋉ H. Since
τh acts on H by conjugation in the same way as the automorphism τch acts, it follows
from 2.1.2 that there exists a group isomorphism β : 〈τch〉H → 〈τ〉H such that
β|H = id and (τch)β = τh. Thus, for the proof of (+), we can replace τ by τch and
may assume that σ α = τ. In this case, (+) follows from 2.1.3.
Lemma 2.9.5. Let G ∼ = C2 × S4 and T ∈ Syl2(G). Suppose S is a group containing
T as a subgroup of index 2. If Op(G) and Z(G) are not normal in S, then the amalgam
(S, G, T, id, id) is uniquely determined up to isomorphism.
Proof. Set Q = Op(G) and C = Z(G). Note that S = S/Z(T ) is not abelian,
since Q is not normal in S. Moreover, S has order 8 and T is elementary abelian of
order 4. Hence, S is dihedral and there exists u ∈ S\T such that u is an involution.
Then X := Z(T )〈u〉 has order 8 and is non-abelian since, by assumption, [C, u] �= 1.
Moreover, Z(T ) is a fours group. Hence, X is dihedral and there is an involution
t ∈ X\Z(T ). Note that X ∩ T = Z(T ) and thus t �∈ T . So we have S = 〈t〉 ⋉ T .
Now by 2.4.6 and 2.9.4, A is uniquely determined up to isomorphism.
Lemma 2.9.6. Let F = GF (4), Q = F 2 , M = SL2(4) and G = M ⋉ Q, where we
consider the natural action of M on Q. Let T ∈ Syl2(G), H = NG(T ) and let X be a
2-closed group containing H as a subgroup of index 2. Assume Q is not normal in X.
Then the amalgam (X, G, H, id, id) is uniquely determined up to isomorphism.
39