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 -