GroupTheory/IsCyclicSylowGroup - Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : GroupTheory/IsCyclicSylowGroup

GroupTheory

  

IsCyclicSylowGroup

  

determine whether a group has cyclic Sylow subgroups

  

IsAbelianSylowGroup

  

determine whether a group has Abelian Sylow subgroups

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

IsCyclicSylowGroup( G )

IsAbelianSylowGroup( G )

Parameters

G

-

a permutation group

Description

• 

A finite group G is a cyclic Sylow group if each of its Sylow subgroups is cyclic. These are often referred to as Z-groups in the literature. Examples of Z-groups include groups of square-free order as well as, of course, every cyclic group. Every such group is supersoluble.

• 

The IsCyclicSylowGroup( G ) command returns the value true if each Sylow subgroup of G is cyclic; otherwise, it returns false.

• 

A finite group G is an Abelian Sylow group if each of its Sylow subgroups is Abelian. Such a group is most often referred to as an A-group. Examples of A-groups include all Abelian groups and all finite groups of cube-free order.

• 

The IsAbelianSylowGroup( G ) command returns true if the Sylow subgroups of G are all Abelian, and returns false otherwise.

• 

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

Examples

withGroupTheory:

The smallest (non-Abelian) Abelian Sylow groups that are not cyclic Sylow groups are the alternating and dihedral groups of order 12.

IsCyclicSylowGroupAlt4

false

(1)

IsAbelianSylowGroupAlt4

true

(2)

IsAbelianSylowGroupPSL2,8

true

(3)

IsCyclicSylowGroupDihedralGroup6

false

(4)

Compatibility

• 

The GroupTheory[IsCyclicSylowGroup] and GroupTheory[IsAbelianSylowGroup] commands were introduced in Maple 2019.

• 

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

See Also

GroupTheory

GroupTheory[IsAbelian]

GroupTheory[IsCyclic]

GroupTheory[SearchSmallGroups]

GroupTheory[SylowSubgroup]