Baby Monster group
Encyclopedia
In the mathematical field of group theory
, the Baby Monster group B (or just Baby Monster) is a group
of order
It is a simple group
, meaning it does not have any normal subgroup
s except for the subgroup consisting only of the identity element, and B itself.
The Baby Monster group is one of the sporadic group
s, 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 smallest faithful matrix representation
of the Baby Monster is of size 4370 over the finite field
of order 2.
The existence of this group was suggested by Bernd Fischer
in unpublished work in the early 1970s during his investigation of {3,4}-transposition groups: groups generated by a class of transpositions such that the product of any two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 actual construction of the Baby Monster was later realized by Jeffrey Leon and Charles Sims
. The name "Baby Monster" was suggested by John Horton Conway
constructed a vertex operator algebra
acted on by the baby monster,
found the maximal subgroups of the baby monster.
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 showed that it does have such an invariant algebra structure if it is reduced modulo 2.
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the Baby Monster group B (or just Baby Monster) is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
- 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47
- = 4154781481226426191177580544000000
- ≈ 4 · 1033.
It is a simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
, meaning it does not have any normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
s except for the subgroup consisting only of the identity element, and B itself.
The Baby Monster group is one of the sporadic group
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
s, and has the second highest order of these, with the highest order being that of the Monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
. The double cover of the Baby Monster is the centralizer of an element of order 2 in the Monster group.
The smallest faithful matrix representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of the Baby Monster is of size 4370 over the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
of order 2.
The existence of this group was suggested by Bernd Fischer
Bernd Fischer (mathematician)
Bernd Fischer is a German mathematician.He is best known to his contributions to the classification of finite simple groups, and he discovered several of the sporadic groups: he introduced 3-transposition groups and constructed the three Fischer groups, described the Baby Monster and computed its...
in unpublished work in 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
Character table
In group theory, a character table is a two-dimensional table whose rows correspond to irreducible group representations, and whose columns correspond to classes of group elements...
; the actual construction of the Baby Monster was later realized by Jeffrey Leon and Charles Sims
Charles Sims (mathematician)
Charles Coffin Sims is an American mathematician best known for his work in group theory. Together with Donald G. Higman he discovered the Higman–Sims group, one of the sporadic groups...
. The name "Baby Monster" was suggested by John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
constructed a vertex operator algebra
Vertex operator algebra
In mathematics, a vertex operator algebra is an algebraic structure that plays an important role in conformal field theory and related areas of physics...
acted on by the baby monster,
found the maximal subgroups of the baby monster.
In characteristic 0 the 4371-dimensional representation of the baby monster does not have a nontrivial invariant algebra structure analogous to the Griess algebra
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 M as its automorphism group. It is named after mathematician R. L. Griess, who constructed it in 1980 and subsequently used it in 1982 to construct M...
, but showed that it does have such an invariant algebra structure if it is reduced modulo 2.