In the area of modern algebra known as
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the baby monster group ''B'' (or, more simply, the baby monster) is a
sporadic simple group of
order
: 2
413
135
67
211131719233147
: = 4154781481226426191177580544000000
: = 4,154,781,481,226,426,191,177,580,544,000,000
: ≈ 4.
''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
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
246320597611213317192329314147 ...
. The
double cover of the baby monster is the
centralizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
of an element of order 2 in the monster group. The
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
is trivial and the
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \oper ...
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Bernd Fischer in unpublished work from the early 1970s during his investigation of -transposition groups: groups generated by a class of transpositions such that the product of any two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 In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of character ...
. 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 Charles Sims may refer to:
* Charles Sims (painter) (1873–1928), British painter
* Charles Sims (mathematician) (1938–2017), American mathematician
* Charles Sims (aviator) (1899–1929), British World War I flying ace
* Charles Sims (American ...
.
Robert Griess
Robert Louis Griess, Jr. (born 1945, Savannah, Georgia) is a mathematician working on finite simple groups and vertex algebras. He is currently the John Griggs Thompson Distinguished University Professor of mathematics at University of Michigan.
...
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
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
.
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 In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fisc ...
, but 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
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of order 2.
constructed a
vertex operator algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usef ...
acted on by the baby monster.
Generalized monstrous moonshine
Conway and Norton suggested in their 1979 paper that
monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979.
...
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''
2, the relevant McKay–Thompson series is
where one can set the constant term .
:
and ''η''(''τ'') is the
Dedekind eta function
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 t ...
.
Maximal subgroups
found the 30 conjugacy classes of maximal subgroups of ''B'' as follows:
* 2.
2E
6(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.
* 2
1+22.Co
2
* Fi
23
* 2
9+16.S
8(2)
* Th
* (2
2 × F
4(2)):2
* 2
2+10+20.(M
22:2 × S
3)
*
30">30L
5(2)
* S
3 × Fi
22:2
*
35">35(S
5 × L
3(2))
* HN:2
* O
8+(3):S
4
* 3
1+8.2
1+6.U
4(2).2
* (3
2:D
8 × U
4(3).2.2).2
* 5:4 × HS:2
* S
4 ×
2F
4(2)
*
11">11(S
4 × 2S
4)
* S
5 × M
22:2
* (S
6 × L
3(4):2).2
* 5
3.L
3(5)
* 5
1+4.2
1+4.A
5.4
* (S
6 × S
6).4
* 5
2:4S
4 × S
5
* L
2(49).2
3
* L
2(31)
* M
11
* L
3(3)
* L
2(17):2
* L
2(11):2
* 47:23
References
*
*
*
*
External links
MathWorld: Baby monster groupAtlas of Finite Group Representations: Baby Monster group
{{DEFAULTSORT:Baby Monster Group
Sporadic groups