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GroupTheory

  

SylowTower

  

construct a Sylow tower for a finite group

  

IsSylowTowerGroup

  

determine if a group is soluble

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

SylowTower( G )

IsSylowTowerGroup( G )

Parameters

G

-

a permutation group

Description

• 

A Sylow tower for a finite group G is a normal series

1=G0G1Gr=G

  

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.

Examples

withGroupTheory:

GAlt4

GA4

(1)

TSylowTowerG

T1,32,4,1,42,3A4

(2)

typeT,NormalSeries

true

(3)

seqGroupOrderH,H=T

1,4,12

(4)

seqIndexTi1,Ti,i=2..:-numelemsT

4,3

(5)

ComplexionT

2,3

(6)

IsSylowTowerGroupDihedralGroup5

true

(7)

Compatibility

• 

The GroupTheory[SylowTower] and GroupTheory[IsSylowTowerGroup] commands were introduced in Maple 2019.

• 

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

See Also

GroupTheory

GroupTheory[OrderedSylowTower]

GroupTheory[PermutationGroup]

GroupTheory[Series]

GroupTheory[SylowSubgroup]