SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

In order to show Theorem 2 we apply Theorem 1 and a pushing-up result, which
was shown by Baumann in [5], Niles in [17] and later improved by Stellmacher in [21].
Apart from that, our proof is self contained. If S is a 2-group then Theorem 2 leads to
the following complete classification.
Theorem 3. Let F be minimal and p = 2. Let q be as in Theorem 1. Then q ≤ 4.
Moreover F ∼ = FS(G) for some finite group G such that S ∈ Syl2(G) and one of the
following holds:
(a) q = 2, S is dihedral of order at least 16 and G = P GL2(r) for some odd prime
power r.
(b) q = 2, S is semidihedral of order at least 16 and for some odd prime power
r, G is the unique extension of L2(r 2 ) of degree 2 with semidihedral Sylow 2-
subgroups.
(c) q = 2, |S| = 2 5 and G = Aut(A6).
(d) q = 4, |S| = 2 7 and G is the semidirect product of L3(4) with the contragredient
automorphism.
In order to deduce Theorem 3 from Theorem 2 we classify the amalgam A realizing
F. The arguments used there are probably not entirely new. However, in order to keep
things reasonably self-contained we prove everything directly as it is often difficult or
impossible to find suitable references.
The groups listed in Theorem 3 all occur in Aschbacher’s theorem about fusion
systems of characteristic 2-type. We are not sure at the moment how our assumptions
are related to Aschbacher’s hypothesis of local characteristic 2. There might be a way
to obtain our result as a corollary of Aschbacher’s result, although our approach still
would have the advantage of avoiding any kind of K-group hypothesis.
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