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GroupTheory

  

FrattiniSeries

  

construct the Frattini series of a group

  

FrattiniLength

  

return the Frattini length of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

FrattiniSeries( G )

FrattiniLength( G )

Parameters

G

-

a permutation group

Description

• 

The Frattini series of a group G is the descending normal series of G whose terms are the successive Frattini subgroups, defined as follows. Let G0=G and, for 0<k, define Gk=ΦGk1. The sequence

G=G0G1Gr

  

of distinct terms is called the Frattini series of G. The number r is called the Frattini length of G.

• 

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

• 

Since the group G is required to be finite, the Frattini series always terminates in the trivial subgroup.

• 

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

Examples

withGroupTheory&colon;

GDihedralGroup8&colon;

fsFrattiniSeriesG

fsD8ΦD8ΦΦD8ΦΦΦD8

(1)

typefs&comma;NormalSeries

true

(2)

FrattiniLengthG

3

(3)

:-numelemsfs

4

(4)

GFrobeniusGroup18&comma;1

G2&comma;93&comma;64&comma;85&comma;7&comma;1&comma;2&comma;4&comma;3&comma;5&comma;7&comma;6&comma;8&comma;9&comma;1&comma;3&comma;62&comma;5&comma;84&comma;7&comma;9

(5)

fsFrattiniSeriesG

fs2&comma;93&comma;64&comma;85&comma;7&comma;1&comma;2&comma;4&comma;3&comma;5&comma;7&comma;6&comma;8&comma;9&comma;1&comma;3&comma;62&comma;5&comma;84&comma;7&comma;9Φ2&comma;93&comma;64&comma;85&comma;7&comma;1&comma;2&comma;4&comma;3&comma;5&comma;7&comma;6&comma;8&comma;9&comma;1&comma;3&comma;62&comma;5&comma;84&comma;7&comma;9ΦΦ2&comma;93&comma;64&comma;85&comma;7&comma;1&comma;2&comma;4&comma;3&comma;5&comma;7&comma;6&comma;8&comma;9&comma;1&comma;3&comma;62&comma;5&comma;84&comma;7&comma;9

(6)

FrattiniLengthG

2

(7)

Compatibility

• 

The GroupTheory[FrattiniSeries] and GroupTheory[FrattiniLength] commands were introduced in Maple 2019.

• 

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

See Also

GroupTheory

GroupTheory[DihedralGroup]

GroupTheory[FrattiniSubgroup]

GroupTheory[FrobeniusGroup]

GroupTheory[LowerPCentralSeries]

GroupTheory[Series]