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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The main result of this work is an explicit construction of -local subgroups of the Monster, the largest sporadic simple group. The groups constructed are the normalizers in the Monster of certain subgroups of order , , and and have shapes

To show that the groups we construct are contained in the Monster, we make use of certain lattices , defined in terms of the code . One step in demonstrating this is to show that the centralizer of an element of order in is contained in the centralizer of an element of order 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 .

**[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*, 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**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*, J. Algebra**118**(1988), 140-149. MR**961332 (89k:94071)****[16]**J. H. Lindsey,*A correlation between*,*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*,*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*, 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)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
20D08,
20N05,
94B60

Retrieve articles in all journals with MSC: 20D08, 20N05, 94B60

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1266532-4

Keywords:
Monster group,
loops

Article copyright:
© Copyright 1995
American Mathematical Society