construct a Sylow tower for a finite group
determine if a group is soluble
OrderedSylowTower( G, complexion = gamma )
OrderedSylowTower( G )
IsOrderedSylowTowerGroup( G, complexion = gamma )
IsOrderedSylowTowerGroup( G )
a permutation group
list of primes including all the prime divisors of the order of G
An ordered Sylow tower of complexion γ 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, and occur in the same order as they do in the list gamma.
A finite group may, or may not, have an ordered Sylow tower of a specified complexion. If it has an ordered Sylow tower of one complexion, it may not have an ordered Sylow tower for a different complexion.
Every finite nilpotent group has a Sylow tower (of every possible complexion), and a finite group with a Sylow tower (of any complexion) is necessarily soluble.
An ordered Sylow tower group is, of course, a Sylow tower group. (See SylowTower.)
The OrderedSylowTower( G, 'complexion' = gamma ) command computes an ordered Sylow tower of compexion gamma 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 IsOrderedSylowTowerGroup( G, 'complexion' = gamma ) command returns true if G has a Sylow tower of complexion gamma, and returns false if not.
Both OrderedSylowTower and IsOrderedSylowTowerGroup can be called without the complex = gamma option, in which case the default complexion used is the list of all the prime divisors of the order of the group G in descending order. (An ordered Sylow tower of this complexion is sometimes called an ordered Sylow tower of supersoluble type.)
G ≔ Alt⁡4
Error, group GroupTheory:-AlternatingGroup(4) has no ordered Sylow tower of complexion [3, 2]
T ≔ OrderedSylowTower⁡G,'complexion'=2,3
The GroupTheory[OrderedSylowTower] and GroupTheory[IsOrderedSylowTowerGroup] commands were introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
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