TY - JOUR

T1 - A characterization of saturated fusion systems over abelian 2-groups

AU - Henke, Ellen

N1 - Acknowledgments
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.

PY - 2014/6/1

Y1 - 2014/6/1

N2 - 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.

AB - 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.

KW - saturated fusion systems

KW - modular representation theory

KW - finite groups

U2 - 10.1016/j.aim.2014.02.020

DO - 10.1016/j.aim.2014.02.020

M3 - Article

VL - 257

SP - 1

EP - 5

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -