HaradaNortonGroupHN

HaradaNortonGroupHN[]

represents the sporadic simple HaradaNorton group .

Details

Background & Context

  • HaradaNortonGroupHN[] represents the HaradaNorton group , which is a group of order TemplateBox[{2, 14}, Superscript].TemplateBox[{3, 6}, Superscript].TemplateBox[{5, 6}, Superscript].7.11.19. It is one of the 26 sporadic simple groups of finite order.
  • The HaradaNorton group is the ninth largest of the sporadic finite simple groups. It was introduced independently by mathematicians Simon Norton and Koichiro Harada in the late 1970s and centralizes an element of order five in the monster group. It thus acts on a 133-dimensional algebra having a commutative but nonassociative product over the field . Along with the other sporadic simple groups, played a foundational role in the monumental (and complete) classification of finite simple groups.
  • The usual group theoretic functions may be applied to HaradaNortonGroupHN[], including GroupOrder, GroupGenerators, GroupElements and so on. However, while HaradaNortonGroupHN[] 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 HaradaNorton group are available via FiniteGroupData["HaradaNorton","prop"].
  • HaradaNortonGroupHN is related to a number of other symbols. HaradaNortonGroupHN is one of the eight groups (along with FischerGroupFi22, FischerGroupFi23, FischerGroupFi24Prime, HeldGroupHe, ThompsonGroupTh, BabyMonsterGroupB and MonsterGroupM) collectively referred to as the "third generation" of sporadic finite simple groups. It is also 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 HaradaNorton group :

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See Also

PermutationGroup

Tutorials

Introduced in 2010
(8.0)