SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

(a) A is an offender on V .
(b) If |V/CV (A)| ≥ |A/CA(V )|, then V CA(V ) ∈ A(G). In particular, we have
A(CG(V )) ⊆ A(G) and J(CG(V )) ≤ J(G).
Proof. Since V CA(V ) is an elementary abelian subgroup of T , we have |V CA(V )| ≤
|A|. Hence,
|V/CV (A)| ≤ |V/V ∩ A| = |V/V ∩ CA(V )| = |V CA(V )/CA(V )| ≤ |A/CA(V )|
Therefore A is an offender on V . If |V/CV (A)| = |A/CA(V )|, then we have equality
above and in particular, |V CA(V )| = |A|. Hence (b) holds.
Lemma 3.2.3 also shows that in the case |V/CV (A)| = |A/CA(V )| we are often able
to obtain some extra information. This motivates the following definition.
Definition 3.2.4. An offender A on V is called an over-offender on V , if we have
|V/CV (A)| < |A/CA(V )|.
Lemma 3.2.5. Let V, W be normal elementary abelian p-subgroups of G with V ≤ W
and [V, J(G)] �= 1. Let A ∈ A(T ) such that [V, A] �= 1, and ACG(W ) is a minimal
with respect to inclusion element of the set
{BCG(W ) : B ∈ J(G), [W, B] �= 1}.
Assume A is not an over-offender on V . Then we have |W/CW (A)| = |A/CA(W )| =
|V/CV (A)| and W = V CW (A).
Proof. It follows from 3.2.3 that |V/CV (A)| = |A/CA(V )|, B := CA(V )V ∈ A(T )
and |W/CW (A)| ≤ |A/CA(W )|. Since [V, A] �= 1 and [V, B] = 1, BCG(W ) is a
proper subset of ACG(W ). Hence, the minimality of ACG(W ) yields [B, W ] = 1.
Thus CA(V ) = CA(W ). It follows that
|W/CW (A)| ≤ |A/CA(W )| = |A/CA(V )|
= |V/CV (A)| = |V CW (A)/CW (A)| ≤ |W/CW (A)|.
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