Glauberman type implies ZJ-functor controls fusion

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group of Glauberman type for a prime) must also satisfy the second group property (i.e., group in which the ZJ-functor controls fusion)
View all group property implications | View all group property non-implications
Get more facts about group of Glauberman type for a prime|Get more facts about group in which the ZJ-functor controls fusion

Statement

Suppose $p$ is a prime number and $G$ is a finite group such that $G$ is of Glauberman type for $p$. Then, the ZJ-functor for the prime $p$ controls fusion in $G$.

Facts used

1. Conjugacy functor whose normalizer generates whole group with p'-core controls fusion

Proof

The proof follows directly from Fact (1) and the definition of a group of Glauberman type as one where the normalizer of the ZJ-functor generates the whole group when taken alongside the p'-core.

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 282, Theorem 4.1, Chapter 8 (p-constrained and p-stable groups), ^{More info}