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.

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

$\mathrm{IsCyclicSylowGroup}\left(\mathrm{Alt}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsAbelianSylowGroup}\left(\mathrm{Alt}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsCyclicSylowGroup}\left(\mathrm{DihedralGroup}\left(6\right)\right)$

$\mathrm{false}$

$\mathrm{IsAbelianSylowGroup}\left(\mathrm{DihedralGroup}\left(6\right)\right)$

$\mathrm{true}$

$\mathrm{IsAbelianSylowGroup}\left(\mathrm{PSL}\left(2\,8\right)\right)$

$\mathrm{true}$

$\mathrm{IsAbelianSylowGroup}\left(\mathrm{JankoGroup}\left(1\right)\right)$

$\mathrm{true}$

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

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