MonsterGroupM

MonsterGroupM[]

represents the sporadic simple monster group .

Details

  • No permutation representation is implemented for MonsterGroupM[].

Background & Context

  • MonsterGroupM[] represents the monster group , which is a group of order TemplateBox[{2, 46}, Superscript].TemplateBox[{3, 20}, Superscript].TemplateBox[{5, 9}, Superscript].TemplateBox[{7, 6}, Superscript].TemplateBox[{11, 2}, Superscript].TemplateBox[{13, 3}, Superscript].17.19.23.29.31.41.47.59.71. It is one of the 26 sporadic simple groups of finite order.
  • The monster group is the largest of the sporadic finite simple groups. It was postulated to exist by mathematician Bernd Fischer in the early 1970s as a simple group containing the baby monster group as the centralizer of an involution. Despite considerable work on the monster group by many mathematicians throughout the 1970s, it was not until Griess explicitly constructed in the early 1980s that its existence was confirmed. The monster group has a faithful linear representation of dimension over the field of two elements, as well as a faithful permutation representation on points. It is a Galois group over the rational numbers, a Hurwitz group and the automorphism group of the so-called monster vertex algebra. Along with the other sporadic simple groups, the monster group played a foundational role in the monumental (and complete) classification of finite simple groups.
  • The usual group theoretic functions may be applied to MonsterGroupM[], including GroupOrder, GroupGenerators, GroupElements and so on. However, while MonsterGroupM[] is a permutation group, due its large order, an explicit permutation representation is impractical for direct implementation. As a result, a number of such group theoretic functions may return unevaluated when applied to it. A number of precomputed properties of the monster group are available via FiniteGroupData["Monster","prop"].
  • MonsterGroupM is related to a number of other symbols. MonsterGroupM is one of the eight groups (along with FischerGroupFi22, FischerGroupFi23, FischerGroupFi24Prime, HeldGroupHe, HaradaNortonGroupHN, BabyMonsterGroupB and ThompsonGroupTh) collectively referred to as the "third generation" of sporadic finite simple groups. MonsterGroupM contains all but six other sporadic groups (the so-called "pariahs") as subquotients.

Examples

Basic Examples  (1)

Order of the monster group :

In[1]:=
Click for copyable input
Out[1]=

See Also

PermutationGroup  BabyMonsterGroupB

Tutorials

Introduced in 2010
(8.0)