GroupTheory - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : GroupTheory/IsLagrangian

GroupTheory

  

IsLagrangian

  

attempt to determine whether a group is Lagrangian

  

IsGCLTGroup

  

determine whether a group is a GCLT group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

IsLagrangian( G )

IsGCLTGroup( G )

Parameters

G

-

a finite group

Description

• 

A finite group G is Lagrangian (or, a CLT-group) if it satisfies the converse of Lagrange's Theorem in the sense that it has a subgroup of order equal to every divisor of its order.

• 

Every finite nilpotent group is Lagrangian, and a finite group is supersoluble if, and only if, each of its subgroups is Lagrangian. (Finite nilpotent groups have a much stronger property: a finite group is nilpotent if, and only if, it has a normal subgroup of order d, for each divisor d of its order.)

• 

The class of Lagrangian groups is neither subgroup- nor quotient-closed.

• 

The IsLagrangian( G ) command attempts to determine whether the group G is Lagrangian.  It returns true if G is Lagrangian and returns false otherwise.

• 

A GCLT-group is a finite group G such that, for each subgroup H of G, and for each prime divisor p of the index [G:H] of H in G, there is a subgroup L of G, containing H, for which the index [L:H] is equal to p. GCLT-groups are most commonly referred to as 𝒥-groups in the literature.

• 

Every GCLT-group is Lagrangian, but not conversely.

• 

The IsGCLTGroup( G ) command attempts to determine whether the group G is a GCLT-group. It returns true if G is a GCLT-group, and returns the value false otherwise.

• 

The group G must be an instance of a permutation group.

Examples

withGroupTheory:

The following examples illustrate that the class of Lagrangian groups is not subgroup-closed.

IsLagrangianSymm4

true

(1)

IsLagrangianAlt4

false

(2)

IsGCLTGroupSymm4

false

(3)

IsGCLTGroupDihedralGroup6

true

(4)

The smallest Lagrangian group that is not a GCLT-group is the direct product of a cyclic group of order 3 and the symmetric group of degree 3.

GPermutationGroupDirectProductCyclicGroup3,Symm3

G1,2,3,4,5,4,5,6

(5)

IsLagrangianG

true

(6)

IsGCLTGroupG

false

(7)

Compatibility

• 

The GroupTheory[IsLagrangian] and GroupTheory[IsGCLTGroup] commands were introduced in Maple 2019.

• 

For more information on Maple 2019 changes, see Updates in Maple 2019.

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[DihedralGroup]

GroupTheory[GroupOrder]

GroupTheory[IsSupersoluble]

GroupTheory[SymmetricGroup]