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.
Theorem 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.
- saturated fusion systems
- modular representation theory
- finite groups