SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

If there is no over-offender in G on V , then we have equality above and A is not an
over-offender on V .
The concept of an offender as introduced above has the drawback that an offender on
a given module is not necessarily an offender on any submodule. However, it turns out
that every submodule of a given FF-module is again an FF-module. Moreover, there is
a special type of offenders which can be inherited to submodules.
Definition 3.2.8. We call a subgroup A of G a best offender on V if
• A/CA(V ) is non-trivial and elementary abelian,
• |A/CA(V )||CV (A)| ≥ |A ∗ /CA ∗(V )||CV (A ∗ )| for all subgroups A ∗ of A.
We write OG(V ) for the set of all best offenders in G on V and JG(V ) for the subgroup
generated by all best offenders in G on V .
Remark 3.2.9. Every best offender on V is an offender on V .
Proof. Let A ≤ G be a best offender on V . Then we have for A ∗ = 1, |V | =
|A ∗ /CA ∗(V )||CV (A ∗ )| ≤ |A/CA(V )||CV (A)|. Hence, A is an offender on V .
Not every offender is a best offender. However, we have
Lemma 3.2.10. The module V is an FF-module iff there is a best offender on V .
Proof. By 3.2.9, it is sufficient to show that there exists a best offender on V if V
is an FF-module. Let A be an offender on V . Choose B to be a subgroup of A
with [V, B] �= 1 for which |B/CB(V )||CV (B)| is maximal. Then by this choice,
|B/CB(V )||CV (B)| ≥ |B ∗ /CB ∗(V )||CV (B ∗ )| for every subgroup B ∗ of B for which
[B ∗ , V ] �= 1. Now let B ∗ be a subgroup of B centralizing V . Again by the maximality
of |B/CB(V )||CV (B)|,
|B/CB(V )||CV (B)| ≥ |A/CA(V )||CV (A)| ≥ |V | = |B ∗ /CB ∗(V )||CV (B ∗ )|.
Hence B is a best offender.
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