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GroupTheory

  

IsSemiprimitive

  

determine whether a permutation group is semi-primitive

  

IsQuasiprimitive

  

determine whether a permutation group is quasi-primitive

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

IsSemiprimitive( G, domain )

IsQuasiprimitive( G, domain )

Parameters

G

-

: PermutationGroup : a permutation group

domain

-

: set(posint) : (optional) a G-invariant subset of the support of G

Description

• 

A permutation group G is quasi-primitive if each of its non-trivial normal subgroups is transitive.

• 

A permutation group G is semi-primitive if each of its non-trivial normal subgroups either is transitive or semi-regular.

• 

Because every non-trivial normal subgroup of a primitive permutation group is transitive, it is clear that semi-primitivity and quasi-primitivity are generalizations of primitivity. In particular, every primitive permutation group is both semi-primitive and quasi-primitive. It also follows from the definitions that a quasi-primitive permutation group is semi-primitive.

• 

The IsQuasiprimitive( G ) command returns true if the permutation group G is quasi-primitive, and returns the value false otherwise.

• 

The IsSemiprimitive( G ) command returns true if the permutation group G is semi-primitive, and returns false if it is not.

• 

The optional domain argument, which must be a G-invariant set, can be used to specify a particular domain of action for G. By default, domain is equal to the support of G, that is, the set of points displaced by some element of G.

Examples

withGroupTheory:

Since symmetric groups are primitive, they are also both semi-primitive and quasi-primitive.

GSymm4

GS4

(1)

IsSemiprimitiveG

true

(2)

IsQuasiprimitiveG

true

(3)

IsPrimitiveG

true

(4)

The cyclic group of order 6 is semi-primitive, but not quasi-primitive.

GCyclicGroup6

GC6

(5)

IsSemiprimitiveG

true

(6)

IsQuasiprimitiveG

false

(7)

IsSemiprimitiveGL2,3

true

(8)

IsQuasiprimitiveGL2,3

false

(9)

andmapIsTransitiveorIsSemiRegular,removeIsTrivial,NormalSubgroupsGL2,3

true

(10)

andmapIsTransitive,removeIsTrivial,NormalSubgroupsGL2,3

false

(11)

GTransitiveGroup24,707:

IsQuasiprimitiveG

true

(12)

IsPrimitiveG

false

(13)

mapIsTransitive,removeIsTrivial,NormalSubgroupsG

true,true

(14)

The following groups fail to be semi-primitive (hence, also quasi-primitive) since they are not even transitive.

GGroupPerm1,2,3,Perm4,5

G1,2,3,4,5

(15)

IsSemiprimitiveG

false

(16)

IsTransitiveG

false

(17)

GCyclicGroup72,'mindegree'

GC72

(18)

orbsmapElements,OrbitsG

orbs1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17

(19)

ARestrictedPermGroupG,orbs1

A1,2,3,4,5,6,7,8

(20)

IsPrimitiveA

false

(21)

IsSemiprimitiveA

true

(22)

IsQuasiprimitiveA

false

(23)

See Also

GroupTheory

GroupTheory[CyclicGroup]

GroupTheory[IsPrimitive]

GroupTheory[IsTransitive]

GroupTheory[Support]