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GroupTheory

  

PermGroupRank

  

compute the rank of a permutation group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

PermGroupRank( G )

Parameters

G

-

a transitive permutation group

Description

• 

The rank of a transitive permutation group G is the number of its sub-orbits, that is, the number of orbits of a point stabiliser. (By convention, the transitivity of an intransitive group is equal to 0.)

• 

The PermGroupRank( G ) command returns the rank of the transitive permutation group G. The group G must be a transitive permutation group. If G is not transitive, an exception is raised.

Examples

withGroupTheory:

The alternating and symmetric groups of degree 4 both have rank 2.

PermGroupRankAlt4

2

(1)

PermGroupRankSymm4

2

(2)

To verify this, compute the number of orbits of the stabiliser of a point.

SStabiliser1,Symm4

S2,4,3,4,2,4,3,2,3,4

(3)

nopsOrbitsS

2

(4)

PermGroupRankFrobeniusGroup100,3

7

(5)

Note that the rank is only defined for transitive permutation groups. If the input to the PermGroupRank command is an intransitive group, then an exception is raised, as in the following example.

GGroupPerm1,2,Perm3,4

G1,2,3,4

(6)

PermGroupRankG

Error, (in PermGroupRank) permutation group must be transitive

IsTransitiveG

false

(7)

Compatibility

• 

The GroupTheory[PermGroupRank] command was introduced in Maple 2020.

• 

For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[FrobeniusGroup]

GroupTheory[IsTransitive]

GroupTheory[Orbits]

GroupTheory[Stabiliser]

GroupTheory[SymmetricGroup]