Baby Monster group
| Algebraic structure → Group theory Group theory |
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Modular groups
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Infinite dimensional Lie group
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In group theory, the Baby Monster group B (or, more simply, the Baby Monster) is a group of order
- 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47
- = 4154781481226426191177580544000000
- ≈ 4 · 1033.
The Baby Monster group is one of the sporadic simple groups, and has the second highest order of these, with the highest order being that of the Monster group. The double cover of the Baby Monster is the centralizer of an element of order 2 in the Monster group.
History
The existence of this group was suggested by Bernd Fischer in unpublished work from the early 1970s during his investigation of {3,4}-transposition groups: groups generated by a class of transpositions such that the product of any two elements has order at most 4, He investigated its properties and computed its character table. The first construction of the Baby Monster was later realized as a permutation group on 13 571 955 000 points using a computer by Jeffrey Leon and Charles Sims.,[1][2] though Griess later found a computer-free construction using the fact that its double cover is contained in the monster. The name "Baby Monster" was suggested by John Horton Conway.[3]
Representations
In characteristic 0 the 4371-dimensional representation of the baby monster does not have a nontrivial invariant algebra structure analogous to the Griess algebra, but Ryba (2007) showed that it does have such an invariant algebra structure if it is reduced modulo 2.
The smallest faithful matrix representation of the Baby Monster is of size 4370 over the finite field of order 2.
Höhn (1996) constructed a vertex operator algebra acted on by the baby monster.
Maximal subgroups
Wilson (1999) gave the 30 classes of maximal subgroups of the baby monster as follows:
2.2E6(2):2 This is the centralizer of an involution, and is the subgroup fixing a point of the smallest permutation representation on 13 571 955 000 points.
21+22.Co2
Fi23
29+16.S8(2)
Th
(22 × F4(2)):2
22+10+20.(M22:2 × S3)
[230].L5(2)
S3 × Fi22:2
[235].(S5 × L3(2))
HN:2
O8+(3):S4
31+8.21+6.U4(2).2
(32:D8 × U4(3).2.2).2
5:4 × HS:2
S4 × 2F4(2)
[311].(S4 × 2S4)
S5 × M22:2
(S6 × L3(4):2).2
53.L3(5)
51+4.21+4.A5.4
(S6 × S6).4
52:4S4 × S5
L2(49).23
L2(31)
M11
L3(3)
L2(17):2
L2(11):2
47:23
References
- ↑ (Gorenstein 1993)
- ↑ Leon, Jeffrey S.; Sims, Charles C. (1977). "The existence and uniqueness of a simple group generated by {3,4}-transpositions". Bull. Amer. Math. Soc. 83 (5): 1039–1040.
- ↑ Ronan, Mark (2006). Symmetry and the Monster. Oxford University Press. pp. 178–179. ISBN 0-19-280722-6.
- Gorenstein, D. (1993), "A brief history of the sporadic simple groups", in Corwin, L.; Gelfand, I. M.; Lepowsky, James, The Gelʹfand Mathematical Seminars, 1990–1992, Boston, MA: Birkhäuser Boston, pp. 137–143, ISBN 978-0-8176-3689-0, MR 1247286
- Höhn, Gerald (1996), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften [Bonn Mathematical Publications], 286, Bonn: Universität Bonn Mathematisches Institut, arXiv:0706.0236, MR 1614941
- Ryba, Alexander J. E. (2007), "A natural invariant algebra for the Baby Monster group", Journal of Group Theory 10 (1): 55–69, doi:10.1515/JGT.2007.006, MR 2288459
- Wilson, Robert A. (1999), "The maximal subgroups of the Baby Monster. I", Journal of Algebra 211 (1): 1–14, doi:10.1006/jabr.1998.7601, MR 1656568