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In the area of modern algebra known as [[group theory]], the '''baby monster group''' ''B'' (or, more simply, the '''baby monster''') is a [[sporadic simple group]] of [[Order (group theory)|order]]

:&nbsp;&nbsp;&nbsp;2<sup>41</sup>{{·}}3<sup>13</sup>{{·}}5<sup>6</sup>{{·}}7<sup>2</sup>{{·}}11{{·}}13{{·}}17{{·}}19{{·}}23{{·}}31{{·}}47
: = 4154781481226426191177580544000000
: = 4,154,781,481,226,426,191,177,580,544,000,000
: ≈ 4{{e|33}}.

''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The existence of this group was suggested by [[Bernd Fischer (mathematician)|Bernd Fischer]] in unpublished work from the early 1970s during his investigation of {3,4}-transposition groups: groups generated by a class of transpositions such that the product of any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 (mathematician)|Charles Sims]],<ref>{{harv|Gorenstein|1993}}</ref><ref>{{Cite journal | last1=Leon | first1=Jeffrey S. | last2=Sims | first2=Charles C. | authorlink2=Charles Sims (mathematician) | title=The existence and uniqueness of a simple group generated by {3,4}-transpositions | journal=Bull. Amer. Math. Soc. | volume=83 | issue=5 | year=1977 | pages=1039&ndash;1040 | url=http://projecteuclid.org/euclid.bams/1183539473 | doi=10.1090/s0002-9904-1977-14369-3}}</ref> though [[Robert 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]].<ref>{{cite book | last = Ronan | first = Mark | title = Symmetry and the monster | publisher = [[Oxford University Press]] | year = 2006 | isbn = 0-19-280722-6 | pages=178&ndash;179 }}</ref>

==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 {{harvtxt|Ryba|2007}} showed that it does have such an invariant algebra structure if it is reduced modulo 2.

The smallest faithful matrix [[group representation|representation]] of the Baby Monster is of size 4370 over the [[finite field]] of order 2.

{{harvtxt|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 ''F''<sub>2</sub>, the relevant McKay–Thompson series is <math>T_{2A}(\tau)</math> where one can set the constant term a(0) = 104.<ref>{{Cite OEIS|1=A007267}}</ref>

:<math>\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+\cdots
\end{align}</math>

and ''η''(''τ'') is the [[Dedekind eta function]].

==Maximal subgroups==
{{harvtxt|Wilson|1999}} found the 30 conjugacy classes of maximal subgroups of ''B'' as follows:

{{div col|colwidth=30em}}
* 2.<SUP>2</SUP>E<SUB>6</SUB>(2):2 {{pad|.5em}} 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. 
* 2<SUP>1+22</SUP>.Co<SUB>2</SUB>
* Fi<SUB>23</SUB>
* 2<SUP>9+16</SUP>.S<SUB>8</SUB>(2)
* Th
* (2<SUP>2</SUP> × F<SUB>4</SUB>(2)):2
* 2<SUP>2+10+20</SUP>.(M<SUB>22</SUB>:2 × S<SUB>3</SUB>)
* [2<SUP>30</SUP>].L<SUB>5</SUB>(2)
* S<SUB>3</SUB> × Fi<SUB>22</SUB>:2
* [2<SUP>35</SUP>].(S<SUB>5</SUB> × L<SUB>3</SUB>(2))
* HN:2
* O<SUB>8</SUB><SUP>+</SUP>(3):S<SUB>4</SUB>
* 3<SUP>1+8</SUP>.2<SUP>1+6</SUP>.U<SUB>4</SUB>(2).2
* (3<SUP>2</SUP>:D<SUB>8</SUB>  × U<SUB>4</SUB>(3).2.2).2
* 5:4 × HS:2
* S<SUB>4</SUB> × <SUP>2</SUP>F<SUB>4</SUB>(2)
* [3<SUP>11</SUP>].(S<SUB>4</SUB> × 2S<SUB>4</SUB>)
* S<SUB>5</SUB> × M<SUB>22</SUB>:2
* (S<SUB>6</SUB> × L<SUB>3</SUB>(4):2).2
* 5<SUP>3</SUP>.L<SUB>3</SUB>(5)
* 5<SUP>1+4</SUP>.2<SUP>1+4</SUP>.A<SUB>5</SUB>.4
* (S<SUB>6</SUB> × S<SUB>6</SUB>).4
* 5<SUP>2</SUP>:4S<SUB>4</SUB> × S<SUB>5</SUB>
* L<SUB>2</SUB>(49).2<SUB>3</SUB>
* L<SUB>2</SUB>(31)
* M<SUB>11</SUB>
* L<SUB>3</SUB>(3)
* L<SUB>2</SUB>(17):2
* L<SUB>2</SUB>(11):2
* 47:23
{{div col end}}

==References==
{{Reflist}}
*{{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | editor1-last=Corwin | editor1-first=L. | editor2-last=Gelfand | editor2-first=I. M. | editor3-last=Lepowsky | editor3-first=James | title=The Gelʹfand Mathematical Seminars, 1990–1992 | url=https://books.google.com/books?id=W1TyAdpZsh8C&pg=PA141&dq=baby+monster+gruppe#v=onepage&q&f=false | publisher=Birkhäuser Boston | location=Boston, MA | isbn=978-0-8176-3689-0 | mr=1247286 | year=1993 | chapter=A brief history of the sporadic simple groups | pages=137–143}}
*{{Citation | last1=Höhn | first1=Gerald | title=Selbstduale Vertexoperatorsuperalgebren und das Babymonster | arxiv=0706.0236 | publisher=Universität Bonn Mathematisches Institut | location=Bonn | series=Bonner Mathematische Schriften [Bonn Mathematical Publications], 286 | mr=1614941 | year=1996| bibcode=2007arXiv0706.0236H }}
*{{Citation | last1=Ryba | first1=Alexander J. E. | title=A natural invariant algebra for the Baby Monster group | doi=10.1515/JGT.2007.006 | mr=2288459 | year=2007 | journal=Journal of Group Theory   | volume=10 | issue=1 | pages=55–69}}
*{{Citation | last1=Wilson | first1=Robert A. | title=The maximal subgroups of the Baby Monster. I | doi=10.1006/jabr.1998.7601 | mr=1656568 | year=1999 | journal=[[Journal of Algebra]]   | volume=211 | issue=1 | pages=1–14}}

==External links==
* [http://mathworld.wolfram.com/BabyMonsterGroup.html MathWorld: Baby monster group]
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/B/ Atlas of Finite Group Representations: Baby Monster group]

{{DEFAULTSORT:Baby Monster Group}}
[[Category:Sporadic groups]]