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Calling Sequence
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OrderedSylowTower( G, complexion = gamma )
OrderedSylowTower( G )
IsOrderedSylowTowerGroup( G, complexion = gamma )
IsOrderedSylowTowerGroup( G )
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Parameters
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G
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a permutation group
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gamma
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list of primes including all the prime divisors of the order of G
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Description
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An ordered Sylow tower of complexion for a finite group is a normal series
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such that, for each , the quotient group is isomorphic to a Sylow -subgroup of , for some prime , and such that , , ..., are all the distinct prime divisors of the order of , and occur in the same order as they do in the list gamma.
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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.
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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.
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An ordered Sylow tower group is, of course, a Sylow tower group. (See SylowTower.)
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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.
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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.
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The IsOrderedSylowTowerGroup( G, 'complexion' = gamma ) command returns true if G has a Sylow tower of complexion gamma, and returns false if not.
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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.)
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Examples
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>
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Compatibility
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The GroupTheory[OrderedSylowTower] and GroupTheory[IsOrderedSylowTowerGroup] commands were introduced in Maple 2019.
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