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GroupTheory

  

LowerPCentralSeries

  

construct the lower p-central series of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

LowerPCentralSeries( p, G )

Parameters

p

-

a prime number

G

-

a permutation group

Description

• 

The lower p-central series of a group G, for a prime number p, 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=GpG,Gk1. The sequence

G=G0G1Gc

  

is called the lower p-central series of G. If the p-residual Gc is the trivial group, then G is a p-group. In this case, the number c is called the p- class of G.

• 

The LowerPCentralSeries( G ) command constructs the lower p-central series of a group G.

• 

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

• 

The lower p-central series of G is 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

G1&comma;2&comma;1&comma;2&comma;34&comma;5

(1)

LowerPCentralSeries2&comma;G

1&comma;2&comma;1&comma;2&comma;34&comma;51&comma;3&comma;2

(2)

LowerPCentralSeries3&comma;G

1&comma;2&comma;1&comma;2&comma;34&comma;5

(3)

LowerPCentralSeries2&comma;QuaternionGroup

Q1&comma;32&comma;45&comma;86&comma;7

(4)

LowerPCentralSeries2&comma;DihedralGroup4

D41&comma;32&comma;4

(5)

LowerPCentralSeries2&comma;DihedralGroup5

D51&comma;3&comma;5&comma;2&comma;4&comma;1&comma;4&comma;2&comma;5&comma;3

(6)

Compatibility

• 

The GroupTheory[LowerPCentralSeries] command was introduced in Maple 2019.

• 

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

See Also

GroupTheory

GroupTheory[DihedralGroup]

GroupTheory[IsPGroup]

GroupTheory[LowerCentralSeries]

GroupTheory[Series]