PreprintarXiv licencearXiv
A positive answer to Barsotti and Carman's question on unit groups of Burnside rings of fusion systems
Abstract
Let F be a saturated fusion system on a finite 2-group S, and let B(F) be the Burnside ring of F. Barsotti and Carman proved that when S is abelian, B(F)^*=B(S)^* if and only if F=F_S(S). Barsotti and Carman asked on page 728 of their paper whether there exists a counterexample to Corollary 9.3 for a nonabelian 2-group. We have given a positive answer. Our main conclusion is that the following are equivalent for all finite 2-groups: (1)F=F_S(S); (2)B(F)^*=B(S)^*; (3)For every maximal subgroup M of S, 1-[S/M] is in B(F)^*.
Related
- Mazurov's question on fixed points of automorphisms of free Burnside groups
- Self-dual sets up to a translation: a negative answer to Milman's question
- A negative answer to the Erdős-Sárkőzy question
- Quadratic bounds for intersecting hypergraphs: An affirmative answer to a question of Duke and Erdős
- A negative answer to a question on tilting objects and two-term complexes
BibTeX
@misc{zhou2026positive,
title = {{A positive answer to Barsotti and Carman's question on unit groups of Burnside rings of fusion systems}},
author = {Chunbo Zhou and Heguo Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07970},
eprint = {2611.07970},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07970},
}
RIS
TY - UNPB
TI - A positive answer to Barsotti and Carman's question on unit groups of Burnside rings of fusion systems
AU - Chunbo Zhou
AU - Heguo Liu
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07970
UR - https://arxiv.org/abs/2611.07970
AB - Let F be a saturated fusion system on a finite 2-group S, and let B(F) be the Burnside ring of F. Barsotti and Carman proved that when S is abelian, B(F)^*=B(S)^* if and only if F=F_S(S). Barsotti and Carman asked on page 728 of their paper whether there exists a counterexample to Corollary 9.3 for a nonabelian 2-group. We have given a positive answer. Our main conclusion is that the following are equivalent for all finite 2-groups: (1)F=F_S(S); (2)B(F)^*=B(S)^*; (3)For every maximal subgroup M of S, 1-[S/M] is in B(F)^*.
ER -