30.01.2013 Views

SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

SHOW MORE
SHOW LESS

Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

We obtain examples of best offenders in a similar way to which we obtained examples<br />

of offenders.<br />

Lemma 3.2.13. Let V be a normal subgroup of G and A ∈ A(G) such that [V, A] �= 1.<br />

Then A ∈ OG(V ).<br />

Proof. See [8, 2.8(e)].<br />

Definition 3.2.14. A subgroup A of G is called a quadratic best offender on V if A is<br />

a best offender on V and [V, A, A] = 1.<br />

Lemma 3.2.15 (Timmesfeld Replacement Theorem I). Every best offender on V con-<br />

tains a subgroup which is a quadratic best offender.<br />

Proof. This is [15, 9.2.3].<br />

The next lemma is a version of the Timmesfeld Replace Theorem for elementary<br />

abelian subgroups of maximal order.<br />

Lemma 3.2.16 (Timmesfeld Replacement Theorem II). Let G be a p-group and V an<br />

elementary abelian normal subgroup of G. Let A ∈ A(G) such that [V, A] �= 1. Then<br />

there exists B ∈ A(ACG(V )) such that [V, B] �= 1 and [V, B, B] = 1.<br />

Proof. Note that by 3.2.13, A ∈ OG(V ). Hence by [15, 9.2.1],<br />

|A/CA(V )| |CV (A)| = |A ∗ /CA ∗(V )| |CV (A ∗ )| for A ∗ = CA([V, A]).<br />

Also, CA ∗(V ) = CA(V ) and hence,<br />

|A| = |A ∗ | · |CV (A ∗ )|<br />

|CV (A)| .<br />

Since V ∩ A ∗ ≤ CV (A), it follows now for B := A ∗ CV (A ∗ ) that<br />

|B| = |A∗ | |CV (A ∗ )|<br />

|V ∩ A ∗ |<br />

≥ |A|.<br />

Note also that B is elementary abelian. Hence, B ∈ A(ACG(V )). By the definition of<br />

B, [V, B, B] = 1. It follows from [15, 9.2.3] that [V, B] = [V, A ∗ ] �= 1.<br />

49

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!