ThompsonGroupTh

ThompsonGroupTh[]

represents the sporadic simple Thompson group .

Details

Background & Context

  • ThompsonGroupTh[] represents the Thompson group , which is a group of order TemplateBox[{2, 15}, Superscript].TemplateBox[{3, 10}, Superscript].TemplateBox[{5, 3}, Superscript].TemplateBox[{7, 2}, Superscript].13.19.31. It is one of the 26 sporadic simple groups of finite order.
  • The Thompson group is the seventh largest of the sporadic finite simple groups. It was discovered by mathematician John Thompson and explicitly constructed by Geoff Smith in the mid-1970s. ThompsonGroupTh was first defined as the automorphism group of a certain lattice in a specific 248-dimensional Lie algebra. In addition, comes with a group action on a vertex operator algebra over the field of three elements, the existence of which stems from the fact that the centralizer of an element of order 3 in the monster group is the product of ThompsonGroupTh and CyclicGroup[3]. Along with the other sporadic simple groups, the Thompson group played a foundational role in the monumental (and complete) classification of finite simple groups.
  • The usual group theoretic functions may be applied to ThompsonGroupTh[], including GroupOrder, GroupGenerators, GroupElements and so on. However, while ThompsonGroupTh[] 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 Thompson group are available via FiniteGroupData["Thompson","prop"].
  • ThompsonGroupTh is related to a number of other symbols. ThompsonGroupTh is one of the eight groups (along with FischerGroupFi22, FischerGroupFi23, FischerGroupFi24Prime, HeldGroupHe, HaradaNortonGroupHN, BabyMonsterGroupB and MonsterGroupM) collectively referred to as the "third generation" of sporadic finite simple groups. It is one of 20 so-called "happy" sporadic groups, which all appear as a subquotient of the monster group.

Examples

Basic Examples  (1)

Order of the Thompson group :

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See Also

PermutationGroup

Tutorials

Introduced in 2010
(8.0)