A characterization of saturated fusion systems over abelian 2-groups

Ellen Henke

doi:10.1016/j.aim.2014.02.020

A characterization of saturated fusion systems over abelian 2-groups

Ellen Henke

Mathematical Science

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

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
Pages (from-to)	1-5
Number of pages	5
Journal	Advances in Mathematics
Volume	257
Early online date	4 Mar 2014
DOIs	https://doi.org/10.1016/j.aim.2014.02.020
Publication status	Published - 1 Jun 2014

Bibliographical note

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.

Keywords

saturated fusion systems
modular representation theory
finite groups

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10.1016/j.aim.2014.02.020Licence: CC BY

http://arxiv.org/abs/1305.4139

Cite this

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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.

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A characterization of saturated fusion systems over abelian 2-groups

Abstract

Bibliographical note

Keywords

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