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GroupTheory

  

DerivedSeries

  

construct the derived series of a group

  

LowerFittingSeries

  

construct the lower Fitting series of a group

  

DerivedLength

  

return the derived length of a group

  

FittingLength

  

return the Fitting length of a group

  

NilpotentLength

  

return the nilpotent length of a group

  

IsSoluble

  

determine if a group is soluble

  

SolubleResidual

  

find the soluble residual of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

DerivedSeries( G )

DerivedLength( G )

LowerFittingSeries( G )

FittingLength( G )

NilpotentLength( G )

IsSoluble( G )

IsSolvable( G )

SolubleResidual( G )

SolvableResidual( G )

Parameters

G

-

a permutation group

Description

• 

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

G=G0G1Gr

  

of distinct terms is called the derived series of G. The number r is called the derived length of G, and the soluble residual Gr of G is the last term of the derived series. If the soluble residual Gr is the trivial group, then we say that G is soluble (or solvable).

• 

The DerivedSeries( G ) command constructs the derived series of a group G. The group G must be an instance of a permutation group. The derived series of G is represented by a series data structure which admits certain operations common to all series.  See GroupTheory[Series].

• 

The DerivedLength( G ) command returns the derived length of G; that is, the length of the derived series of G. This is the number of subgroup inclusions - so it is one less than the number of groups in the derived series.

• 

The IsSoluble( G ) command (or IsSolvable( G ), as an alias) returns true if the group G is soluble, and returns the value false otherwise.

• 

The SolubleResidual( G ) command (or SolvableResidual( G ), as an alias) returns the soluble residual Gr of G. It can also be applied to a derived series object.

• 

The lower Fitting series of a group G is the descending normal series of G whose terms are the successive nilpotent residuals. The sequence

G=G0G1Gr

  

of distinct terms is called the lower Fitting series of G if Gi+1 is defined to be the nilpotent residual of Gi. Then G is soluble if, and only if, this series reaches the trivial subgroup. Its length is called the Fitting length (also known as the nilpotent length) of G.

• 

Note that the Fitting length of G is equal to 1 precisely when G is nilpotent as, in this case, the nilpotent residual of G is trivial.

• 

The LowerFittingSeries( G ) command computes the lower Fitting series of the permutation group G. Like the derived series, the lower Fitting series of G is represented by a series data structure.  Again, see GroupTheory[Series].

• 

The FittingLength( G ) returns the Fitting length of G.  The NilpotentLength( G ) command is identical, and is provided as an alias.

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)

DerivedSeriesG

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

(2)

DerivedLengthG

2

(3)

IsSolubleG

true

(4)

dsDerivedSeriesAlternatingGroup4

dsA41&comma;23&comma;4&comma;1&comma;42&comma;3

(5)

forgindsdoprintgenddo&colon;

A4

1&comma;23&comma;4&comma;1&comma;42&comma;3

(6)

SolubleResidualds

(7)

LowerFittingSeriesAlt4

A41&comma;23&comma;4&comma;1&comma;42&comma;3

(8)

FittingLengthSymm4

3

(9)

FittingLengthDihedralGroup8

1

(10)

Compatibility

• 

The GroupTheory[DerivedSeries], GroupTheory[DerivedLength], GroupTheory[IsSoluble] and GroupTheory[SolubleResidual] commands were introduced in Maple 17.

• 

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

• 

The GroupTheory[LowerFittingSeries], GroupTheory[FittingLength] and GroupTheory[NilpotentLength] commands were introduced in Maple 2019.

• 

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

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[DerivedSubgroup]

GroupTheory[LowerCentralSeries]

GroupTheory[NilpotentResidual]

GroupTheory[PermutationGroup]

GroupTheory[Series]