determine whether a group has cyclic Sylow subgroups
determine whether a group has Abelian Sylow subgroups
IsCyclicSylowGroup( G )
IsAbelianSylowGroup( G )
a permutation group
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.
The smallest (non-Abelian) Abelian Sylow groups that are not cyclic Sylow groups are the alternating and dihedral groups of order 12.
The GroupTheory[IsCyclicSylowGroup] and GroupTheory[IsAbelianSylowGroup] commands were introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
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