GroupTheory/IsCyclicSylowGroup - Maple Help

GroupTheory

 IsCyclicSylowGroup
 determine whether a group has cyclic Sylow subgroups
 IsAbelianSylowGroup
 determine whether a group has Abelian Sylow subgroups

 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

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$

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

 > $\mathrm{IsCyclicSylowGroup}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{false}}$ (1)
 > $\mathrm{IsAbelianSylowGroup}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{IsAbelianSylowGroup}\left(\mathrm{PSL}\left(2,8\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsCyclicSylowGroup}\left(\mathrm{DihedralGroup}\left(6\right)\right)$
 ${\mathrm{false}}$ (4)

Compatibility

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