Geodesic
Equivalences between fusion systems of finite groups of Lie type
Broto, Carles1 ; Møller, Jesper2 ; Oliver, Bob3
1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain
2 Matematisk Institut, Universitetsparken 5, DK–2100 København, Denmark
3 LAGA, Institut Galilée, Av. J-B Clément, F–93430 Villetaneuse, France
Journal of the American Mathematical Society, Tome 25 (2012) no. 1, pp. 1-20

Voir la notice de l'article provenant de la source American Mathematical Society

We prove, for certain pairs $G,G’$ of finite groups of Lie type, that the $p$-fusion systems $\mathcal {F}_p(G)$ and $\mathcal {F}_p(G’)$ are equivalent. In other words, there is an isomorphism between a Sylow $p$-subgroup of $G$ and one of $G’$ which preserves $p$-fusion. This occurs, for example, when $G=\mathbb {G}(q)$ and $G’=\mathbb {G}(q’)$ for a simple Lie “type” $\mathbb {G}$, and $q$ and $q’$ are prime powers, both prime to $p$, which generate the same closed subgroup of $p$-adic units. Our proof uses homotopy-theoretic properties of the $p$-completed classifying spaces of $G$ and $G’$, and we know of no purely algebraic proof of this result.
DOI : 10.1090/S0894-0347-2011-00713-3

Broto, Carles 1 ; Møller, Jesper 2 ; Oliver, Bob 3

1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain
2 Matematisk Institut, Universitetsparken 5, DK–2100 København, Denmark
3 LAGA, Institut Galilée, Av. J-B Clément, F–93430 Villetaneuse, France 
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Broto, Carles; Møller, Jesper; Oliver, Bob. Equivalences between fusion systems of finite groups of Lie type. Journal of the American Mathematical Society, Tome 25 (2012) no. 1, pp. 1-20. doi: 10.1090/S0894-0347-2011-00713-3

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