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Local subgroups of the Monster and odd code loops


Author: Thomas M. Richardson
Journal: Trans. Amer. Math. Soc. 347 (1995), 1453-1531
MSC: Primary 20D08; Secondary 20N05, 94B60
DOI: https://doi.org/10.1090/S0002-9947-1995-1266532-4
MathSciNet review: 1266532
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Abstract: The main result of this work is an explicit construction of $ p$-local subgroups of the Monster, the largest sporadic simple group. The groups constructed are the normalizers in the Monster of certain subgroups of order $ 3^{2}$, $ 5^{2}$, and $ 7^{2}$ and have shapes

$\displaystyle {3^{2 + 5 + 10}}\cdot ({M_{11}} \times GL(2,3)),\quad {5^{2 + 2 + 4}}\cdot {S_3} \times GL(2,5)),\quad {\text{and}}{7^{2 + 1 + 2}}\cdot GL(2,7)$

. These groups result from a general construction which proceeds in three steps. We start with a self-orthogonal code $ C$ of length $ n$ over the field $ {\mathbb{F}_p}$, where $ p$ is an odd prime. The first step is to define a code loop $ L$ whose structure is based on $ C$. The second step is to define a group $ N$ of permutations of functions from $ \mathbb{F}_p^2$ to $ L$. The final step is to show that $ N$ has a normal subgroup $ K$ of order $ {p^2}$. The result of this construction is the quotient group $ N/K$ of shape $ {p^{2 + m + 2m}}(S \times GL(2,p))$, where $ m + 1 = \dim (C)$ and $ S$ is the group of permutations of Aut$ (C)$.

To show that the groups we construct are contained in the Monster, we make use of certain lattices $ \Lambda (C)$, defined in terms of the code $ C$. One step in demonstrating this is to show that the centralizer of an element of order $ p$ in $ N/K$ is contained in the centralizer of an element of order $ p$ in the Monster. The lattices are useful in this regard since a quotient of the automorphism group of the lattice is a composition factor of the appropriate centralizer in the Monster.

This work was inspired by a similar construction using code loops based on binary codes that John Conway used to construct a subgroup of the Monster of shape $ {2^{2 + 11 + 22}}\cdot ({M_{24}} \times GL(2,2))$.


References [Enhancements On Off] (What's this?)

  • [1] H. F. Blichfeldt, Finite collineation groups, Univ. of Chicago Press, Chicago, IL, 1917.
  • [2] R. H. Brack, A survey of binary systems, Springer-Verlag, New York, Heidelberg, and Berlin, 1958. MR 0093552 (20:76)
  • [3] J. H. Conway, A simple construction for the Fischer-Griess Monster group, Invent. Math. 79 (1985), 513-540. MR 782233 (86h:20019)
  • [4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ATLAS of finite groups, Claredon Press, Oxford, 1985. MR 827219 (88g:20025)
  • [5] J. H. Conway and J. A. Sloane, Sphere packings, lattices, and groups, Springer-Verlag, New York, Heidelberg, and Berlin, 1988. MR 920369 (89a:11067)
  • [6] L. Finkelstein and A. Rudvalis, Maximal subgroups of the Hall-Janko-Wales group, J. Algebra 24 (1973), 486-493. MR 0323889 (48:2242)
  • [7] R. L. Griess, Schur multipliers of some sporadic simple groups, J. Algebra 32 (1974), 445-466. MR 0382426 (52:3310)
  • [8] -, The friendly giant, Invent. Math. 62 (1982), 1-102. MR 671653 (84m:20024)
  • [9] -, The Monster and its non-associative algebra, Proceedings of a Conference on Finite Groups (Montreal, 1985), pp. 121-157.
  • [10] -, Code loops and a large finite group containing triality for $ {D_4}$, Atti Convegno Internazionale Teoria dei Gruppi e Geometria Combinatoria, Firenze, 1986, pp. 79-98.
  • [11] -, Code loops, J. Algebra 100 (1986), 224-234. MR 839580 (87i:20124)
  • [12] -, A Moufang loop, the exceptional Jordan algebra, and a cubic form in $ 27$ variables, J. Algebra 131 (1990), 281-293. MR 1055009 (91g:17044)
  • [13] P. M. Johnson, Loops of nilpotence class two, preprint.
  • [14] G. Karpilovsky, The Schur multiplier, Oxford Univ. Press, Oxford, 1986. MR 1200015 (93j:20002)
  • [15] M. Kitazume, Code loops and even codes over $ {\mathbb{F}_4}$, J. Algebra 118 (1988), 140-149. MR 961332 (89k:94071)
  • [16] J. H. Lindsey, A correlation between $ PS{U_4}(3)$, the Suzuki group, and the Conway group, Trans. Amer. Math. Soc. 157 (1971), 189-204. MR 0283097 (44:330)
  • [17] -, A new lattice for the Hall-Janko group, Proc. Amer. Math. Soc. 103 (1988), 703-709. MR 947642 (89g:20075)
  • [18] J. van Lint, An introduction to coding theory, Springer-Verlag, New York, Heidelberg, and Berlin, 1982. MR 658134 (84e:94001)
  • [19] H. Maschke, Math. Ann. 51 (1899), 253-298.
  • [20] J. G. Thompson, Uniqueness of the Fischer-Griess Monster, Bull. London Math. Soc. 11 (1979), 340-346. MR 554400 (81e:20024)
  • [21] J. Tits, Quaternions over $ \mathbb{Q}(\sqrt 5 )$, Leech's lattice and the sporadic group of Hall-Janko, J. Algebra 63 (1980), 56-75. MR 568564 (82k:20034)
  • [22] H. N. Ward, A form for $ {M_{11}}$, J. Algebra 37 (1975), 340-351. MR 0384907 (52:5777)
  • [23] -, Combinatorial polarization, Discrete Math. 26 (1979), 185-197. MR 535244 (80m:05012)
  • [24] R. A. Wilson, Maximal subgroups of the Suzuki group, J. Algebra 84 (1983), 151-188. MR 716777 (86e:20034b)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1266532-4
Keywords: Monster group, loops
Article copyright: © Copyright 1995 American Mathematical Society

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