Equivalences between fusion systems of finite groups of Lie type

Details Equivalences between fusion systems of finite groups of Lie type Equivalences between fusion systems of finite groups of Lie type

2008-03-12

arxiv.org/abs/0803.1746v2

J. Amer. Math. Soc. 25 (2012), 1-20

Mathematics

Group Theory

20 pages, uses diagrams.sty and xy-pic, second version

Scientific paper

10.1090/S0894-0347-2011-00713-3

We prove, for certain pairs G,G of finite groups of Lie type, that the p-fusion systems for G and 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=H(q) and G'=H(q') for a simple Lie type H, and q and q' are prime powers, both prime to p, which generate the same closed subgroup of the 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.

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