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GroupTheory

  

LowerCentralSeries

  

construct the lower central series of a group

  

UpperCentralSeries

  

construct the upper central series of a group

  

IsNilpotent

  

determine if a group is nilpotent

  

NilpotencyClass

  

find the nilpotency class of a group

  

NilpotentResidual

  

find the nilpotency residual of a group

  

Hypercenter

  

find the hypercenter of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

LowerCentralSeries( G )

UpperCentralSeries( G )

IsNilpotent( G )

NilpotencyClass( G )

NilpotentResidual( G )

Hypercenter( G )

Parameters

G

-

a permutation group

Description

• 

The lower central series of a group G is the descending normal series of G whose terms are the successive commutator subgroups, defined as follows. Let G0=G and, for 0<k, define Gk=G&comma;Gk1. The sequence

G=G0G1Gc

  

is called the lower central series of G. If the nilpotent residual Gc is the trivial group, then we say that G is nilpotent. In this case, the number c is called the nilpotency class of G, and the nilpotent residual Gc of G is the last term of the lower central series.

• 

The LowerCentralSeries( G ) command constructs the lower central series of a group G. The group G must be an instance of a permutation group.

• 

The IsNilpotent( G ) command determines whether a group G is nilpotent.

• 

The NilpotencyClass( G ) command returns the nilpotency class of G; that is, the length of the lower central series of G.

• 

The NilpotentResidual( G ) command returns the nilpotent residual of a group G.

• 

The upper central series of a group G is the ascending normal series of G whose terms are defined, recursively, as follows. Let G0=1 and, for 0<k, define Gk to be the pre-image, in G, of the center of the quotient group GGk1.  (Thus, G1 is just the center of G.) The sequence

1=G0G1Gc

  

is called the upper central series of G.

• 

The UpperCentralSeries( G ) command constructs the upper central series of a group G.

• 

The group G is nilpotent if, and only if, the last term Gc of the upper central series is equal to G. In general, the final term Gc is called the hypercenter of G.

• 

The Hypercenter( G ) command returns the hypercenter of a group G.

• 

The Hypercentre command is provided as an alias.

• 

The group G must be an instance of a permutation group.

• 

Both the lower and upper central series of G are represented by a series data structure which admits certain operations common to all series.  See GroupTheory[Series].

Examples

withGroupTheory&colon;

GPermutationGroup1&comma;2&comma;1&comma;2&comma;3&comma;4&comma;5

GroupTheory:-PermutationGroupmodule...end module&comma;module...end module&comma;degree&equals;5

(1)

LowerCentralSeriesG

GroupTheory:-LowerCentralSeriesmodule...end module

(2)

NilpotencyClassG

1

(3)

IsNilpotentG

false

(4)

lcsLowerCentralSeriesAlternatingGroup4

GroupTheory:-LowerCentralSeriesmodule...end module

(5)

forginlcsdoprintgenddo&colon;

GroupTheory:-AlternatingGroup4

GroupTheory:-PermutationGroupmodule...end module&comma;module...end module&comma;module...end module&comma;degree&equals;4&comma;supergroup&equals;GroupTheory:-PermutationGroupmodule...end module&comma;module...end module&comma;degree&equals;4

(6)

IsNilpotentDihedralGroup8

true

(7)

IsNilpotentDihedralGroup12

false

(8)

NilpotentResidualDihedralGroup12

GroupTheory:-PermutationGroupmodule...end module&comma;module...end module&comma;degree&equals;12&comma;supergroup&equals;GroupTheory:-PermutationGroupmodule...end module&comma;module...end module&comma;degree&equals;12&comma;supergroup&equals;GroupTheory:-PermutationGroupmodule...end module&comma;module...end module&comma;degree&equals;12

(9)

UpperCentralSeriesDihedralGroup16

GroupTheory:-UpperCentralSeriesmodule...end module

(10)

UpperCentralSeriesDihedralGroup12

GroupTheory:-UpperCentralSeriesmodule...end module

(11)

HypercenterDihedralGroup12

GroupTheory:-PermutationGroupmodule...end module&comma;module...end module&comma;degree&equals;12&comma;supergroup&equals;GroupTheory:-PermutationGroupmodule...end module&comma;module...end module&comma;degree&equals;12

(12)

IsNilpotentDihedralGroup42kassumingk::&apos;posint&apos;

true

(13)

IsNilpotentDihedralGroup6kassumingk::&apos;posint&apos;

false

(14)

Compatibility

• 

The GroupTheory[LowerCentralSeries] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[DerivedSeries]

GroupTheory[PermutationGroup]

GroupTheory[Series]