construct a Sylow tower for a finite group
determine if a group is soluble
SylowTower( G )
IsSylowTowerGroup( G )
a permutation group
A Sylow tower for a finite group G is a normal series
such that, for each i, the quotient group GiGi+1 is isomorphic to a Sylow pi-subgroup of G, for some prime pi, and such that p1, p2, ..., pr are all the distinct prime divisors of the order of G.
A finite group may, or may not, have a Sylow tower. If such a Sylow tower exists, then the ordered sequence p1,p2,..,pr of primes (or any ordered sequence of prime numbers containing it in the same order) is called the complexion of the Sylow tower.
Every finite nilpotent group has a Sylow tower (of arbitrary complexion), and a finite group with a Sylow tower is necessarily soluble.
A group with a Sylow tower need not have an ordered Sylow tower. (See OrderedSylowTower.)
The SylowTower( G ) command computes a Sylow tower for the group G if one exists. The returned Sylow tower is an object of type NormalSeries.
In addition to the methods available for any Series object, a Sylow tower T also supports the Complexion( T ) method, which returns the complexion of the computed tower, as a list of primes.
The IsSylowTowerGroup( G ) command returns true if G has a Sylow tower (of some complexion), and returns false if no Sylow tower of any complexion exists.
G ≔ Alt⁡4
T ≔ SylowTower⁡G
The GroupTheory[SylowTower] and GroupTheory[IsSylowTowerGroup] commands were introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
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