SuzukiGroupSuz

SuzukiGroupSuz[]

represents the sporadic simple Suzuki group .

Details

  • By default, SuzukiGroupSuz[] is represented as a permutation group acting on points {1,,1782}.

Background & Context

  • SuzukiGroupSuz[] represents the Suzuki group , which is a group of order TemplateBox[{2, 13}, Superscript].TemplateBox[{3, 7}, Superscript].TemplateBox[{5, 2}, Superscript].7.11.13. It is one of the 26 sporadic simple groups of finite order. The default representation of SuzukiGroupSuz is as a permutation group on the symbols having two generators.
  • The Suzuki group is the thirteenth smallest of the sporadic finite simple groups. It was discovered by mathematician Michio Suzuki in the late 1960s. SuzukiGroupSuz is a rank-3 permutation group on the symbols having point stabilizer isomorphic to the group and consisting of the points of the Lie group over the field of four elements. SuzukiGroupSuz is related to the Conway groups in the sense that the latter may be realized as automorphism groups of the so-called (real) Leech lattice, while SuzukiGroupSuz has as its universal cover the automorphism group of the so-called complex Leech lattice. Along with the other sporadic simple groups, the Suzuki group played a foundational role in the monumental (and complete) classification of finite simple groups.
  • The usual group theoretic functions may be applied to SuzukiGroupSuz[], including GroupOrder, GroupGenerators, GroupElements and so on. However, due its large order, a number of such group theoretic functions may return unevaluated when applied to it. A number of precomputed properties of the Suzuki group are available via FiniteGroupData["Suzuki","prop"].
  • SuzukiGroupSuz is related to a number of other symbols. SuzukiGroupSuz is one of the seven groups (along with ConwayGroupCo1, ConwayGroupCo2, ConwayGroupCo3, JankoGroupJ2, HigmanSimsGroupHS and McLaughlinGroupMcL) 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. Note that a class of different groups also named after Suzuki is cataloged in FiniteGroupData via FiniteGroupData[{"SuzukiGroup",n},"prop"].

Examples

Basic Examples  (3)

Order of the group :

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Number of points moved by the generators of a permutation representation of :

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Order of a pseudorandom element of the group :

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See Also

PermutationGroup

Tutorials

Introduced in 2010
(8.0)