Equivalences between fusion systems of finite groups of Lie type
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- by Carles Broto, Jesper M. Møller and Bob Oliver
- J. Amer. Math. Soc. 25 (2012), 1-20
- DOI: https://doi.org/10.1090/S0894-0347-2011-00713-3
- Published electronically: July 8, 2011
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Abstract:
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.References
- K. K. S. Andersen, J. Grodal, J. M. Møller, and A. Viruel, The classification of $p$-compact groups for $p$ odd, Ann. of Math. (2) 167 (2008), no. 1, 95–210. MR 2373153, DOI 10.4007/annals.2008.167.95
- Michael Aschbacher, Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 1986. MR 895134
- Armand Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207 (French). MR 51508, DOI 10.2307/1969728
- A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573, DOI 10.1007/978-3-540-38117-4
- Glen E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin-New York, 1967. MR 0214062, DOI 10.1007/BFb0082690
- Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR 781344, DOI 10.1007/978-3-662-12918-0
- C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Extensions of $p$-local finite groups, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3791–3858. MR 2302515, DOI 10.1090/S0002-9947-07-04225-0
- Carles Broto and Jesper M. Møller, Chevalley $p$-local finite groups, Algebr. Geom. Topol. 7 (2007), 1809–1919. MR 2366180, DOI 10.2140/agt.2007.7.1809
- W. G. Dwyer and C. W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139 (1994), no. 2, 395–442. MR 1274096, DOI 10.2307/2946585
- Paul Fong and R. James Milgram, On the geometry and cohomology of the simple groups $G_2(q)$ and $^3\!D_4(q)$, Group representations: cohomology, group actions and topology (Seattle, WA, 1996) Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 221–244. MR 1603163, DOI 10.1090/pspum/063/1603163
- Eric M. Friedlander, Étale homotopy of simplicial schemes, Annals of Mathematics Studies, No. 104, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 676809
- Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
- Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups. Number 3. Part I. Chapter A, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998. Almost simple $K$-groups. MR 1490581, DOI 10.1090/surv/040.3
- Robert L. Griess Jr. and A. J. E. Ryba, Embeddings of $\textrm {PGL}_2(31)$ and $\textrm {SL}_2(32)$ in $E_8(\mathbf C)$, Duke Math. J. 94 (1998), no. 1, 181–211. With appendices by Michael Larsen and J.-P. Serre. MR 1635916, DOI 10.1215/S0012-7094-98-09409-1
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- Stefan Jackowski, James McClure, and Bob Oliver, Self-homotopy equivalences of classifying spaces of compact connected Lie groups, Fund. Math. 147 (1995), no. 2, 99–126. MR 1341725
- Peter B. Kleidman, The maximal subgroups of the Steinberg triality groups $^3D_4(q)$ and of their automorphism groups, J. Algebra 115 (1988), no. 1, 182–199. MR 937609, DOI 10.1016/0021-8693(88)90290-6
- Peter B. Kleidman, The maximal subgroups of the Chevalley groups $G_2(q)$ with $q$ odd, the Ree groups $^2G_2(q)$, and their automorphism groups, J. Algebra 117 (1988), no. 1, 30–71. MR 955589, DOI 10.1016/0021-8693(88)90239-6
- John Martino and Stewart Priddy, Unstable homotopy classification of $BG_p\sphat$, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 1, 119–137. MR 1356164, DOI 10.1017/S030500410007403X
- R. James Milgram, On the geometry and cohomology of the simple groups $G_2(q)$ and $^3\!D_4(q)$. II, Group representations: cohomology, group actions and topology (Seattle, WA, 1996) Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 397–418. MR 1603199, DOI 10.1090/pspum/063/1603199
- Guido Mislin, On group homomorphisms inducing mod-$p$ cohomology isomorphisms, Comment. Math. Helv. 65 (1990), no. 3, 454–461. MR 1069820, DOI 10.1007/BF02566619
- Jesper M. Møller, $N$-determined 2-compact groups. I, Fund. Math. 195 (2007), no. 1, 11–84. MR 2314074, DOI 10.4064/fm195-1-2
- J. M. Møller and D. Notbohm, Centers and finite coverings of finite loop spaces, J. Reine Angew. Math. 456 (1994), 99–133. MR 1301453
- Bob Oliver, Equivalences of classifying spaces completed at odd primes, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 2, 321–347. MR 2092063, DOI 10.1017/S0305004104007728
- Bob Oliver, Equivalences of classifying spaces completed at the prime two, Mem. Amer. Math. Soc. 180 (2006), no. 848, vi+102. MR 2203209, DOI 10.1090/memo/0848
- Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380, DOI 10.1007/978-1-4684-9458-7
- G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
Bibliographic Information
- Carles Broto
- Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain
- MR Author ID: 42005
- Email: broto@mat.uab.es
- Jesper M. Møller
- Affiliation: Matematisk Institut, Universitetsparken 5, DK–2100 København, Denmark
- ORCID: 0000-0003-4053-2418
- Email: moller@math.ku.dk
- Bob Oliver
- Affiliation: LAGA, Institut Galilée, Av. J-B Clément, F–93430 Villetaneuse, France
- MR Author ID: 191965
- Email: bobol@math.univ-paris13.fr
- Received by editor(s): March 17, 2010
- Received by editor(s) in revised form: June 14, 2011
- Published electronically: July 8, 2011
- Additional Notes: The first author is partially supported by FEDER-MICINN grant MTM 2010-20692
The second author was partially supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation
The third author was partially supported by UMR 7539 of the CNRS, and by project ANR BLAN08-2_338236, HGRT - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc. 25 (2012), 1-20
- MSC (2010): Primary 20D06; Secondary 55R37, 20D20
- DOI: https://doi.org/10.1090/S0894-0347-2011-00713-3
- MathSciNet review: 2833477