SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

We obtain examples of best offenders in a similar way to which we obtained examples
of offenders.
Lemma 3.2.13. Let V be a normal subgroup of G and A ∈ A(G) such that [V, A] �= 1.
Then A ∈ OG(V ).
Proof. See [8, 2.8(e)].
Definition 3.2.14. A subgroup A of G is called a quadratic best offender on V if A is
a best offender on V and [V, A, A] = 1.
Lemma 3.2.15 (Timmesfeld Replacement Theorem I). Every best offender on V con-
tains a subgroup which is a quadratic best offender.
Proof. This is [15, 9.2.3].
The next lemma is a version of the Timmesfeld Replace Theorem for elementary
abelian subgroups of maximal order.
Lemma 3.2.16 (Timmesfeld Replacement Theorem II). Let G be a p-group and V an
elementary abelian normal subgroup of G. Let A ∈ A(G) such that [V, A] �= 1. Then
there exists B ∈ A(ACG(V )) such that [V, B] �= 1 and [V, B, B] = 1.
Proof. Note that by 3.2.13, A ∈ OG(V ). Hence by [15, 9.2.1],
|A/CA(V )| |CV (A)| = |A ∗ /CA ∗(V )| |CV (A ∗ )| for A ∗ = CA([V, A]).
Also, CA ∗(V ) = CA(V ) and hence,
|A| = |A ∗ | · |CV (A ∗ )|
|CV (A)| .
Since V ∩ A ∗ ≤ CV (A), it follows now for B := A ∗ CV (A ∗ ) that
|B| = |A∗ | |CV (A ∗ )|
|V ∩ A ∗ |
≥ |A|.
Note also that B is elementary abelian. Hence, B ∈ A(ACG(V )). By the definition of
B, [V, B, B] = 1. It follows from [15, 9.2.3] that [V, B] = [V, A ∗ ] �= 1.
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