# Baby Monster group

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
= 4,154,781,481,226,426,191,177,580,544,000,000
≈ 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The existence of this group was suggested by Bernd Fischer in unpublished work from the early 1970s during his investigation of {3,4}-transposition groups: groups generated by a class of transpositions such that the product of any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.

## 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 (),

\begin{align}j_{2A}(\tau) &=T_{2A}(\tau)+104\\ &=\Big(\big(\tfrac{\eta(\tau)}{\eta(2\tau)}\big)^{12}+2^6 \big(\tfrac{\eta(2\tau)}{\eta(\tau)}\big)^{12}\Big)^2\\ &=\frac{1}{q} + 104 + 4372q + 96256q^2 +1240002q^3+10698752q^4\dots \end{align}

and η(τ) is the Dedekind eta function. Let $r_n$ = 1, 4371, 96255, 1139374, 9458750, 9550635,...() be the degrees of irreducible representations of B. Then,

\begin{align} 1 &= r_1\\ 4372 &= r_1+r_2\\ 96256 &= r_1+r_3\\ 1240002 &= 2r_1+r_2+r_3+r_4\\ 10698752 &= 2r_1+r_2+r_3+r_4+r_5\\ \end{align}

and so on. Incidentally, like the j-function, the function j2A(τ) appears in pi formulas and can also assume an integer value for appropriate arguments. For example,

$j_{2A}\Big(\tfrac{1}{2}\sqrt{-58}\Big) = 396^4$

which implies,

$e^{\pi\sqrt{58}} = 396^4 - 104.00000017\dots$

and leads to the pi formula known to Ramanujan.

## 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

1. ^
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.