Abstract
We extend the notion of a $p$-local finite group (defined in [BLO03]) to the notion of a $p$-local group. We define morphisms of $p$-local groups, obtaining thereby a category, and we introduce the notion of a representation of a $p$-local group via signalizer functors associated with groups. We construct a chain $\mathfrak{G}=(\mathcal{G}_0\to\mathcal{G}_1\to\cdots)$ of $2$-local finite groups, via a representation of a chain $\frak G^*=(G_0\to G_1\to\cdots)$ of groups, such that $\mathcal{G}_0$ is the $2$-local finite group of the third Conway sporadic group ${\rm Co}_3$, and for $n>0$, $\mathcal{G}_n$ is one of the $2$-local finite groups constructed by Levi and Oliver in [LO02]. We show that the direct limit $\mathcal{G}$ of $\mathfrak{G}$ exists in the category of $2$-local groups, and that it is the $2$-local group of the union of the chain $\mathfrak{G}^*$. The $2$-completed classifying space of $\mathcal{G}$ is shown to be the classifying space $B\operatorname{D}\operatorname{I}(4)$ of the exotic $2$-compact group of Dwyer and Wilkerson [DW93].
-
[A] M. Aschbacher, Finite Group Theory, Cambridge: Cambridge Univ. Press, 1986.
@book {A, MRKEY = {895134},
AUTHOR = {Aschbacher, Michael},
TITLE = {Finite Group Theory},
SERIES = {Cambridge Stud. Adv. Math.},
NUMBER = {10},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1986},
PAGES = {x+274},
ISBN = {0-521-30341-9},
MRCLASS = {20-01 (20B05 20C05 20D05)},
MRNUMBER = {89b:20001},
MRREVIEWER = {Stephen D. Smith},
ZBLNUMBER = {0583.20001},
} -
[B]
D. Benson, "Conway’s group ${ Co}_3$ and the Dickson invariants," Manuscripta Math., vol. 85, iss. 2, pp. 177-193, 1994.
@article {B, MRKEY = {1302871},
AUTHOR = {Benson, Dave},
TITLE = {Conway's group {${\rm Co}\sb 3$} and the {D}ickson invariants},
JOURNAL = {Manuscripta Math.},
FJOURNAL = {Manuscripta Mathematica},
VOLUME = {85},
YEAR = {1994},
NUMBER = {2},
PAGES = {177--193},
ISSN = {0025-2611},
CODEN = {MSMHB2},
MRCLASS = {55R35 (20D08)},
MRNUMBER = {95h:55018},
MRREVIEWER = {Haynes R. Miller},
DOI = {10.1007/BF02568192},
ZBLNUMBER = {0853.55018},
} -
[BCGLO]
C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, "Subgroup families controlling $p$-local finite groups," Proc. London Math. Soc., vol. 91, iss. 2, pp. 325-354, 2005.
@article {BCGLO, MRKEY = {2167090},
AUTHOR = {Broto, Carles and Castellana, Nat{à}lia and Grodal, Jesper and Levi, Ran and Oliver, Bob},
TITLE = {Subgroup families controlling {$p$}-local finite groups},
JOURNAL = {Proc. London Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
VOLUME = {91},
YEAR = {2005},
NUMBER = {2},
PAGES = {325--354},
ISSN = {0024-6115},
CODEN = {PLMTAL},
MRCLASS = {20J99 (20D15 20D20 55R35)},
MRNUMBER = {2007e:20111},
MRREVIEWER = {David J. Green},
DOI = {10.1112/S0024611505015327},
ZBLNUMBER = {1090.20026},
} -
[BLO1]
C. Broto, R. Levi, and B. Oliver, "The homotopy theory of fusion systems," J. Amer. Math. Soc., vol. 16, iss. 4, pp. 779-856, 2003.
@article {BLO1, MRKEY = {1992826},
AUTHOR = {Broto, Carles and Levi, Ran and Oliver, Bob},
TITLE = {The homotopy theory of fusion systems},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {16},
YEAR = {2003},
NUMBER = {4},
PAGES = {779--856},
ISSN = {0894-0347},
MRCLASS = {55R35 (20D15 55R40)},
MRNUMBER = {2004k:55016},
MRREVIEWER = {David J. Green},
DOI = {10.1090/S0894-0347-03-00434-X},
ZBLNUMBER = {1033.55010},
} -
[BLO2] C. Broto, R. Levi, and B. Oliver, Discrete models for the $p$-local homotopy theory of compact Lie groups, 2005.
@misc{BLO2,
author = {Broto, Carles and Levi, Ran and Oliver, Bob},
TITLE = {Discrete models for the $p$-local homotopy theory of compact Lie groups},
YEAR = {2005},
ZBLNUMBER = {1135.55008},
} -
[C] C. Chevalley, The Algebraic Theory of Spinors and Clifford Algebras, New York: Springer-Verlag, 1997.
@book {C, MRKEY = {1636473},
AUTHOR = {Chevalley, Claude},
TITLE = {The Algebraic Theory of Spinors and {C}lifford Algebras},
SERIES={Collected Works, {\rm Vol}. 2},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1997},
PAGES = {xiv+214},
ISBN = {3-540-57063-2},
MRCLASS = {01A75 (15A66 81R25)},
MRNUMBER = {99f:01028},
ZBLNUMBER = {0899.01032},
} -
[COS] A. Chermak, B. Oliver, and S. Shpectorov, The simple connectivity of ${ Sol}(q)$, preprint, 2006.
@misc{COS,
author = {Chermak, A and Oliver, B and Shpectorov, S.},
TITLE = {The simple connectivity of ${\rm Sol}(q)$, preprint},
YEAR = {2006},
} -
[D] W. G. Dwyer, "Classifying spaces and homology decompositions," in Homotopy Theoretic Methods in Group Cohomology, Dwyer, W. G. and Hans-Werner, H., Eds., Basel: Birkhäuser Verlag, 2001, pp. 3-53.
@incollection{D,
author = {Dwyer, W. G.},
TITLE = {Classifying spaces and homology decompositions},
BOOKTITLE={Homotopy Theoretic Methods in Group Cohomology},
SERIES={Advanced Courses in Math.\/ {\rm CRM} Barcelona},
EDITOR={Dwyer, W.G. and Hans-Werner, H.},
PUBLISHER={Birkhäuser Verlag},
ADDRESS={ Basel},
YEAR = {2001},
PAGES={3--53},
} -
[DW]
W. G. Dwyer and C. W. Wilkerson, "A new finite loop space at the prime two," J. Amer. Math. Soc., vol. 6, iss. 1, pp. 37-64, 1993.
@article {DW, MRKEY = {1161306},
AUTHOR = {Dwyer, W. G. and Wilkerson, C. W.},
TITLE = {A new finite loop space at the prime two},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {6},
YEAR = {1993},
NUMBER = {1},
PAGES = {37--64},
ISSN = {0894-0347},
MRCLASS = {55P35 (55P99)},
MRNUMBER = {93d:55011},
MRREVIEWER = {John F. Oprea},
DOI = {10.2307/2152794},
ZBLNUMBER = {0769.55007},
} -
[F]
L. Finkelstein, "The maximal subgroups of Conway’s group $C_{3}$ and McLaughlin’s group," J. Algebra, vol. 25, pp. 58-89, 1973.
@article {F, MRKEY = {0346046},
AUTHOR = {Finkelstein, Larry},
TITLE = {The maximal subgroups of {C}onway's group {$C\sb{3}$} and {M}c{L}aughlin's group},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {25},
YEAR = {1973},
PAGES = {58--89},
ISSN = {0021-8693},
MRCLASS = {20D05},
MRNUMBER = {49 \#10772},
MRREVIEWER = {T. M. Gagen},
DOI = {10.1016/0021-8693(73)90075-6},
ZBLNUMBER = {0263.20010},
} -
[G1]
D. M. Goldschmidt, "A conjugation family for finite groups," J. Algebra, vol. 16, pp. 138-142, 1970.
@article {G1, MRKEY = {0260869},
AUTHOR = {Goldschmidt, David M.},
TITLE = {A conjugation family for finite groups},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {16},
YEAR = {1970},
PAGES = {138--142},
ISSN = {0021-8693},
MRCLASS = {20.40},
MRNUMBER = {41 \#5489},
MRREVIEWER = {J. L. Alperin},
DOI = {10.1016/0021-8693(70)90046-3},
ZBLNUMBER = {0198.04306},
} -
[G2]
D. M. Goldschmidt, "Automorphisms of trivalent graphs," Ann. of Math., vol. 111, iss. 2, pp. 377-406, 1980.
@article {G2, MRKEY = {569075},
AUTHOR = {Goldschmidt, David M.},
TITLE = {Automorphisms of trivalent graphs},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {111},
YEAR = {1980},
NUMBER = {2},
PAGES = {377--406},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {05C25 (20B25 20E06)},
MRNUMBER = {82a:05052},
MRREVIEWER = {Gareth A. Jones},
DOI = {10.2307/1971203},
ZBLNUMBER = {0475.05043},
} -
[GLS] D. Gorenstein, R. Lyons, and R. Solomon, The classification of the finite simple groups, Providence, RI: Amer. Math. Soc., 1994.
@book {GLS, MRKEY = {1303592},
AUTHOR = {Gorenstein, Daniel and Lyons, Richard and Solomon, Ronald},
TITLE = {The classification of the finite simple groups},
SERIES = {Math. Surveys and Monogr.},
NUMBER = {40},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1994},
PAGES = {xiv+165},
ISBN = {0-8218-0334-4},
MRCLASS = {20D05 (20-02)},
MRNUMBER = {95m:20014},
MRREVIEWER = {J. L. Alperin},
ZBLNUMBER={1069.20011},
} -
[H] A. Hatcher, Algebraic topology, Cambridge: Cambridge Univ. Press, 2002.
@book {H, MRKEY = {1867354},
AUTHOR = {Hatcher, Allen},
TITLE = {Algebraic topology},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2002},
PAGES = {xii+544},
ISBN = {0-521-79160-X; 0-521-79540-0},
MRCLASS = {55-01 (55-00)},
MRNUMBER = {2002k:55001},
MRREVIEWER = {Donald W. Kahn},
ZBLNUMBER = {1044.55001},
} -
[J] N. Jacobson, Basic Algebra. II, San Francisco, CA: W. H. Freeman and Co., 1980.
@book {J, MRKEY = {571884},
AUTHOR = {Jacobson, Nathan},
TITLE = {Basic Algebra. {\rm II}},
PUBLISHER = {W. H. Freeman and Co.},
ADDRESS = {San Francisco, CA},
YEAR = {1980},
PAGES = {xix+666},
ISBN = {0-7167-1079-X},
MRCLASS = {00A05},
MRNUMBER = {81g:00001},
MRREVIEWER = {M. F. Smiley},
ZBLNUMBER={0441.16001},
} -
[K]
W. M. Kantor, "Some exceptional $2$-adic buildings," J. Algebra, vol. 92, iss. 1, pp. 208-223, 1985.
@article {K, MRKEY = {772480},
AUTHOR = {Kantor, William M.},
TITLE = {Some exceptional {$2$}-adic buildings},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {92},
YEAR = {1985},
NUMBER = {1},
PAGES = {208--223},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {51B05 (20G25)},
MRNUMBER = {86j:51007},
MRREVIEWER = {Francis Buekenhout},
DOI = {10.1016/0021-8693(85)90155-3},
ZBLNUMBER = {0562.51006},
} -
[KS] H. Kurzweil and B. Stellmacher, The Theory of Finite Groups. An Introduction, New York: Universitext Springer-Verlag, 2004.
@book {KS, MRKEY = {2014408},
AUTHOR = {Kurzweil, Hans and Stellmacher, Bernd},
TITLE = {The Theory of Finite Groups. An Introduction},
PUBLISHER = {{\it Universitext} Springer-Verlag},
ADDRESS = {New York},
YEAR = {2004},
PAGES = {xii+387},
ISBN = {0-387-40510-0},
MRCLASS = {20-01 (20Dxx)},
MRNUMBER = {2004h:20001},
ZBLNUMBER = {1047.20011},
} -
[LS] I. Leary and R. Stancu, Realizing fusion systems, preprint, 2006.
@misc{LS,
author = {Leary, I. and Stancu, R},
TITLE = {Realizing fusion systems, preprint, 2006},
} -
[LO]
R. Levi and B. Oliver, "Construction of 2-local finite groups of a type studied by Solomon and Benson," Geom. Topol., vol. 6, pp. 917-990, 2002.
@article {LO, MRKEY = {1943386},
AUTHOR = {Levi, Ran and Oliver, Bob},
TITLE = {Construction of 2-local finite groups of a type studied by {S}olomon and {B}enson},
JOURNAL = {Geom. Topol.},
FJOURNAL = {Geometry and Topology},
VOLUME = {6},
YEAR = {2002},
PAGES = {917--990},
ISSN = {1465-3060},
MRCLASS = {55R35 (20D99 55R37)},
MRNUMBER = {2003k:55016},
MRREVIEWER = {Jesper M. Møller},
DOI = {10.2140/gt.2002.6.917},
ZBLNUMBER = {1021.55010},
} -
@article {P, MRKEY = {2253665},
AUTHOR = {Puig, Lluis},
TITLE = {Frobenius categories},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {303},
YEAR = {2006},
NUMBER = {1},
PAGES = {309--357},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {20C20 (20D15)},
MRNUMBER = {2007j:20011},
MRREVIEWER = {Alexander Zimmermann},
DOI = {10.1016/j.jalgebra.2006.01.023},
ZBLNUMBER = {1110.20011},
} -
[R]
G. R. Robinson, "Amalgams, blocks, weights, fusion systems and finite simple groups," J. Algebra, vol. 314, iss. 2, pp. 912-923, 2007.
@article {R, MRKEY = {2344591},
AUTHOR = {Robinson, Geoffrey R.},
TITLE = {Amalgams, blocks, weights, fusion systems and finite simple groups},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {314},
YEAR = {2007},
NUMBER = {2},
PAGES = {912--923},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {20E06 (20D15 20D20)},
MRNUMBER = {2008f:20046},
MRREVIEWER = {H. Heineken},
DOI = {10.1016/j.jalgebra.2007.05.010},
ZBLNUMBER = {05201709},
} -
[Se] J. Serre, Trees, New York: Springer-Verlag, 1980.
@book {Se, MRKEY = {607504},
AUTHOR = {Serre, Jean-Pierre},
TITLE = {Trees},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1980},
PAGES = {ix+142},
ISBN = {3-540-10103-9},
MRCLASS = {20H10 (05C05 22E50)},
MRNUMBER = {82c:20083},
ZBLNUMBER = {1013.20001},
} -
[So]
R. Solomon, "Finite groups with Sylow $2$-subgroups of type $3$," J. Algebra, vol. 28, pp. 182-198, 1974.
@article {So, MRKEY = {0344338},
AUTHOR = {Solomon, Ronald},
TITLE = {Finite groups with {S}ylow {$2$}-subgroups of type {$3$}},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {28},
YEAR = {1974},
PAGES = {182--198},
ISSN = {0021-8693},
MRCLASS = {20D20},
MRNUMBER = {49 \#9077},
MRREVIEWER = {Mark P. Hale, Jr.},
DOI = {10.1016/0021-8693(74)90031-3},
ZBLNUMBER = {0293.20022},
} -
[St] R. Steinberg, Lecture Notes on Chevalley Groups, Yale Univ. Lecture Notes, New Haven, CT, 1967.
@misc{St,
author = {Steinberg, R.},
TITLE = {{\it Lecture Notes on Chevalley Groups},
{\it Yale Univ. Lecture Notes},
New Haven, CT, 1967},
} -
[T] J. Tits, "Sur le groupe des automorphismes d’un arbre," in Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), New York: Springer-Verlag, 1970, pp. 188-211.
@incollection {T, MRKEY = {0299534},
AUTHOR = {Tits, Jacques},
TITLE = {Sur le groupe des automorphismes d'un arbre},
BOOKTITLE = {Essays on Topology and Related Topics ({M}émoires dédiés à {G}eorges de {R}ham)},
PAGES = {188--211},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1970},
MRCLASS = {05C25},
MRNUMBER = {45 \#8582},
MRREVIEWER = {A. Mowshowitz},
ZBLNUMBER = {0214.51301},
} -
[W] B. A. F. Wehrfritz, Infinite Linear Groups. An Account of the Group-Theoretic Properties of Infinite Groups of Matrices, New York: Springer-Verlag, 1973, vol. 76.
@book {W, MRKEY = {0335656},
AUTHOR = {Wehrfritz, B. A. F.},
TITLE = {Infinite Linear Groups. {A}n Account of the Group-Theoretic Properties of Infinite Groups of Matrices},
SERIES = {Ergeb. Mat. Grensg.},
VOLUME={76},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1973},
PAGES = {xiv+229},
MRCLASS = {20HXX},
MRNUMBER = {49 \#436},
MRREVIEWER = {I. Stewart},
ZBLNUMBER = {0261.20038},
}