JankoGroupJ2

JankoGroupJ2[]

represents the sporadic simple Janko group .

Details

  • By default, JankoGroupJ2[] is represented as a permutation group acting on points {1,,100}.
  • This group is also known as the HallJanko group or HallJankoWales group.

Background & Context

  • JankoGroupJ2[] represents the Janko group , which is a group of order TemplateBox[{2, 7}, Superscript].TemplateBox[{3, 3}, Superscript].TemplateBox[{5, 2}, Superscript].7. It is one of the 26 sporadic simple groups of finite order. The default representation of JankoGroupJ2 is as a permutation group on the symbols having two generators. may also be referred to as the HallJanko group (denoted ) or the HallJankoWales group.
  • The Janko group is the fifth smallest of the sporadic finite simple groups. It was discovered (along with JankoGroupJ1 and JankoGroupJ3) by mathematician Zvonimir Janko in the mid 1900s, making these groups tied for second in chronological order of discovery among sporadic groups. JankoGroupJ2 is the unique Janko group occurring as a subquotient of the monster group. In addition to its permutation representation, can be defined in terms of generators and relations as . It is contained in the Conway group , thus making it the only Janko group to be part of the "second generation" of sporadic simple groups. Along with the other sporadic simple groups, the Janko groups played a foundational role in the monumental (and complete) classification of finite simple groups.
  • The usual group theoretic functions may be applied to JankoGroupJ2[], including GroupOrder, GroupGenerators, GroupElements and so on. A number of precomputed properties of the Janko group are available via FiniteGroupData[{"Janko",2},"prop"].
  • JankoGroupJ2 is related to a number of other symbols. JankoGroupJ2 is one of the seven groups (along with the Conway groups ConwayGroupCo1, ConwayGroupCo2 and ConwayGroupCo3, and together with HigmanSimsGroupHS, McLaughlinGroupMcL and SuzukiGroupSuz) collectively referred to as the "second 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 group :

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Generators of a permutation representation of the group :

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See Also

PermutationGroup  JankoGroupJ1  JankoGroupJ3  JankoGroupJ4

Tutorials

Introduced in 2010
(8.0)