Abstract
Given a saturated fusion system F over a 2-group S, we prove that S is abelian provided any element of S is F-conjugate to an element of Z(S). This generalizes a Theorem of Camina--Herzog, leading to a significant simplification of its proof. More importantly, it follows that any 2-block B of a finite group has abelian defect groups if all B-subsections are major. Furthermore, every 2-block with a symmetric stable center has abelian defect groups.
Original language | English |
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Pages (from-to) | 1-5 |
Number of pages | 5 |
Journal | Advances in Mathematics |
Volume | 257 |
Early online date | 4 Mar 2014 |
DOIs | |
Publication status | Published - 1 Jun 2014 |
Bibliographical note
AcknowledgmentsTheorem 1 and Corollary 2 were conjectured by Külshammer, Navarro, Sambale and Tiep and answer the question posed in [5, Question 3.1] for p=2. The author would like to thank Benjamin Sambale for drawing her attention to this problem. Furthermore, she would like to thank an anonymous referee for suggesting Corollary 3.
Keywords
- saturated fusion systems
- modular representation theory
- finite groups