SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

in studying p-blocks of finite groups. Later these categories became known as satu-
rated fusion systems. The now standard notation and terminology was introduced by
Broto, Levi and Oliver in [7]. We define these in Chapter 5.
A fusion system F on an arbitrary finite p-group S is a category whose objects are
all subgroups of S and whose morphisms are group monomorphisms. Among these
morphisms are all conjugations by elements of S. For example if S is a subgroup of a
(not necessarily finite) group H then we can form a fusion system FS(H) on S where
the morphisms are just the maps induced by conjugation with elements of H.
In saturated fusion systems some extra properties hold which mimic the features
of Sylow subgroups in finite groups. So the fusion systems FS(G) where G is a finite
group and S a Sylow p-subgroup of G provide the main examples of saturated fusion
systems. However, there are saturated fusion systems which cannot be obtained in this
way. These fusion systems are called exotic.
From now on, for the remainder of the introduction, fusion systems are assumed to
be saturated and F is supposed to be a fusion systems on a finite p-group S.
Many exotic examples have been discovered so far. At first the search was some
what random. Later more systematic approaches were used which then led to the dis-
covery of new exotic examples. For instance, Ruiz and Viruel classified in [19] all
fusion systems on extraspecial groups of order p 3 and exponent p, and found that some
of these, for p = 3, 5, 7 and 13, were exotic.
Anyway, this choice of a p-group seems to us rather random as in fact any other
choice might be. Therefore we believe it is now time to investigate classes of fusion
systems which can be described in the fusion theoretic language rather than just in
terms of the underlying groups. Naturally we also expect this to lead to the discovery
of new exotic examples, as well as to the development of more structure theory for
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