Baby monster group

Last updated April 12, 2024

In the area of modern algebra known as group theory, the baby monster groupB (or, more simply, the baby monster) is a sporadic simple group of order

B is one of the 26 sporadic 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. The outer automorphism group is trivial and the Schur multiplier has order 2.

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The existence of this group was suggested by Bernd Fischer in unpublished work from the early 1970s during his investigation of {3,4}-transposition groups: groups generated by a class of transpositions such that the product of any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] Robert Griess later found a computer-free construction using the fact that its double cover is contained in the monster group. 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.

Generalized monstrous moonshine

Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For the Baby monster B or F₂, the relevant McKay–Thompson series is $T_{2A}(\tau )$ where one can set the constant term a(0) = 104.^[4]

{\begin{aligned}j_{2A}(\tau )&=T_{2A}(\tau )+104\\&=\left(\left({\tfrac {\eta (\tau )}{\eta (2\tau )}}\right)^{12}+2^{6}\left({\tfrac {\eta (2\tau )}{\eta (\tau )}}\right)^{12}\right)^{2}\\&={\frac {1}{q}}+104+4372q+96256q^{2}+1240002q^{3}+10698752q^{4}+\cdots \end{aligned}}

and η(τ) is the Dedekind eta function.

Maximal subgroups

Wilson (1999) found the 30 conjugacy classes of maximal subgroups of B 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
2^2+10+20.(M₂₂:2 × S₃)
[2³⁰].L₅(2)
S₃ × Fi₂₂:2
[2³⁵].(S₅ × L₃(2))
HN:2
O⁺
₈(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

Related Research Articles

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group $defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .$

In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
= 2⁴⁶ · 3²⁰ · 5⁹ · 7⁶ · 11² · 13³ · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
≈ 8×10⁵³.

In the mathematical classification of finite simple groups, there are 26 or 27 groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups.

In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co₁, Co₂ and Co₃ along with the related finite group Co₀ introduced by (Conway 1968, 1969).

In the area of modern algebra known as group theory, the Higman–Sims group HS is a sporadic simple group of order

In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory.

In the area of modern algebra known as group theory, the Thompson groupTh is a sporadic simple group of order

In the area of modern algebra known as group theory, the Held groupHe is a sporadic simple group of order

In the area of abstract algebra known as group theory, the O'Nan groupO'N or O'Nan–Sims group is a sporadic simple group of order

In the area of modern algebra known as group theory, the Rudvalis groupRu is a sporadic simple group of order

In the area of modern algebra known as group theory, the Harada–Norton groupHN is a sporadic simple group of order

In the area of modern algebra known as group theory, the Mathieu groupM₁₂ is a sporadic simple group of order

In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order

In mathematics, the modular lambda function λ(τ) is a highly symmetric Holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2). Over any point τ, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve $, where the map is defined as the quotient by the [-1] involution.$

In mathematics, the Weber modular functions are a family of three functions f, f₁, and f₂, studied by Heinrich Martin Weber.

In the area of modern algebra known as group theory, the Fischer groupFi₂₄ or F₂₄′ or F₃₊ is a sporadic simple group of order

In the area of modern algebra known as group theory, the Fischer groupFi₂₃ is a sporadic simple group of order

In the area of modern algebra known as group theory, the Fischer groupFi₂₂ is a sporadic simple group of order

In the area of modern algebra known as group theory, the Conway group is a sporadic simple group of order

In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as,

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. doi: 10.1090/s0002-9904-1977-14369-3 .
↑ Ronan, Mark (2006). Symmetry and the monster . Oxford University Press. pp. 178–179. ISBN 0-19-280722-6.
↑ Sloane, N. J. A. (ed.). "SequenceA007267". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.

Gorenstein, D. (1993), "A brief history of the sporadic simple groups", in Corwin, L.; Gelfand, I. M.; Lepowsky, James (eds.), 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 , Bibcode:2007arXiv0706.0236H, 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, S2CID 122359097
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

External links

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] ( Gorenstein 1993 )

[2] 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. doi: 10.1090/s0002-9904-1977-14369-3 .

[3] Ronan, Mark (2006). Symmetry and the monster . Oxford University Press. pp. 178–179. ISBN 0-19-280722-6.

[4] Sloane, N. J. A. (ed.). "SequenceA007267". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.

[1]

[2]

[3]

[4]