SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

Lemma 2.3.5. Let G be a finite group and S a normal p-subgroup of G such that
CG(S) ≤ S. Assume there is Q ≤ S with CS(Q) ≤ Q. Then CG(Q) is a p-group. In
particular, if S ∈ Sylp(G) then CG(Q) ≤ Q.
Proof. Let R be a p-prime subgroup of CG(Q). Then A = R × Q acts on the p-group
S. Moreover, CS(Q) ≤ Q ≤ CS(R). Therefore, by Thompson’s P × Q-lemma,
[S, R] = 1. Hence, R ≤ CG(S) ≤ S and R = 1. This shows that CG(Q) is a p-group.
If S ∈ Sylp(G), we get now CG(Q) ≤ CS(Q) ≤ Q.
2.4 Dihedral and semidihedral 2-groups
We remind the reader that the group G is called dihedral if it is generated by two
involutions. Equivalently, there exist elements t, x ∈ G such that t is an involution,
G = 〈t〉 ⋉ 〈x〉 and x t = x −1 . We are particularly interested in the case where G is a
2-group. Note that we also regard fours groups as dihedral groups. A group which is
similar to the dihedral 2-group is the semidihedral 2-group, which is defined as follows.
Definition 2.4.1. G is called semidihedral, if there exist elements x, t ∈ G such that t
is an involution, o(x) = 2 n for some n ≥ 3, G = 〈t〉 ⋉ 〈x〉 and x t = x 2n−1 −1 .
While studying dihedral and semidihedral 2-groups, the automorphisms of cyclic 2-
groups play an important role. We summarize some of their properties in the following
lemma.
Lemma 2.4.2. Let X = 〈x〉 be a cyclic group of order 2 n , n ≥ 2.
(a) xσ2 �= x−1 for every σ ∈ Aut(X).
(b) Aut(X) has order 2 n−1 .
(c) If n ≥ 3 then the automorphisms α, β, γ of X defined by
x α = x −1 , x β = x 2n−1 −1 , x γ = x 2 n−1 +1
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