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GroupTheory

  

JankoGroup

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

JankoGroup( n, opts )

Parameters

n

-

: {1,2,3,4} : integer indicating which Janko group to construct

opts

-

option of the form form = "permgroup" or form = "fpgroup"

Description

• 

The Janko groups J1, J2, J3 and J4 are four of the sporadic simple groups.  The first Janko group J1 was the first sporadic simple group to be discovered (in 1965, by Zvonimir Janko) since the Mathieu groups were discovered in the nineteenth century.  Janko predicted the existence of the remaining Janko groups, which were later proved to exist by others.

• 

Three of the Janko groups (all except J2) are pariahs: they do not occur as subquotients of the Monster. The group J2 is also called the Hall-Janko group or the Hall-Janko-Wales group; its existence was proven by Marshall Hall Jr. and David Wales in 1968. The third Janko group J3 was shown to exist in 1969 by Graham Higman and John McKay. Simon Norton proved that the fourth Janko group J4 exists in 1980.

• 

The JankoGroup( n ) command returns a permutation group, or a finitely presented group, isomorphic to the Janko group Jn, for n = 1, 2, 3, 4.

• 

For n = 1, 2, 3, you may specify either form = "permgroup" (the default) or form = "fpgroup".  For n = 4, you must specify form = "fpgroup". The fourth Janko group J4 is not available as a permutation group in the current version of Maple.

Examples

withGroupTheory:

GJankoGroup2

G:=J2

(1)

DegreeG

100

(2)

GroupOrderG

604800

(3)

IsSimpleG

true

(4)

GJankoGroup4,'form'=fpgroup

G:=J4

(5)

GroupOrderG

86775571046077562880

(6)

RelatorsG

t,t,x,x,y,y,y,1t,1x,t,x,1t,1y,1x,1y,1x,1y,1x,y,1x,y,1x,1y,1x,1y,t,y,x,y,x,1y,x,1y,x,y,x,y,x,y,x,y,x,y,x,1y,x,y,x,y,x,1y,x,y,x,y,x,1y,x,y,x,1y,x,1y,x,y,x,1y,x,1y,x,y,x,1y,x,1y,1x,1y,1x,1y,x,y,x,y,1x,1y,1x,1y,x,y,x,y,1x,1y,1x,1y,x,y,x,y,1x,1y,1x,1y,x,y,x,y,1x,1y,1x,1y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,x,y,1x,1y,x,y,1x,1y,x,y,1x,1y,x,y,1x,1y,x,y,1x,1y,x,y,1x,1y,x,y,1x,1y,x,y,1x,1y,x,y,1x,1y,x,y,1x,1y,x,y,1x,1y,x,y,1x,1y,x,y,y,1x,y,1x,1y,1x,y,1x,1y,t,y,x,1y,x,y,x,1y,x,y,1x,y,1x,1y,1x,y,1x,1y,t,y,x,1y,x,y,x,1y,x,y,1x,y,1x,1y,1x,y,1x,1y,t,y,x,1y,x,y,x,1y,x,x,y,x,y,x,1y,x,y,x,1y,x,y,x,1y,x,y,x,y,x,1y,x,y,x,1y,x,y,x,1y,x,y,x,y,x,1y,x,y,x,1y,x,y,x,1y,x,y,x,y,x,1y,x,y,x,1y,x,y,x,1y,y,x,y,x,y,x,y,y,x,y,x,y,x,y,y,x,y,x,y,x,y,t,1y,1y,1x,1y,1x,1y,1x,1y,1x,1y,1x,1y,1x,1y,1y,1x,1y,1x,1y,1x,t,x,y,x,y,x,y,y,x,y,x,y,x,y,x,y,x,y,x,y,y,y,x,y,x,y,x,y,y,x,y,x,y,x,y,y,x,y,x,y,x,y,t,1y,1y,1x,1y,1x,1y,1x,1y,1x,1y,1x,1y,1x,1y,1y,1x,1y,1x,1y,1x,t,x,y,x,y,x,y,y,x,y,x,y,x,y,x,y,x,y,x,y,y

(7)

Compatibility

• 

The GroupTheory[JankoGroup] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

See Also

GroupTheory[Degree]

GroupTheory[GroupOrder]

GroupTheory[IsSimple]

GroupTheory[Relators]

 


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