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GroupTheory

  

CompositionSeries

  

construct a composition series of a finite group

  

CompositionLength

  

compute the composition length of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

CompositionSeries( G )

CompositionLength( G )

Parameters

G

-

a permutation group

Description

• 

A composition series of a group G is a subnormal series

G=G0G1Gr=1

of G, for which each term is a maximal normal subgroup in the preceding term, so that the successive quotients GkGk+1 are simple groups.

• 

Every finite group has a composition series, and any two composition series for a finite group have the same number of terms, and the multi-set of isomorphism types of the quotients GkGk+1 is unique (apart from order).  The number r of terms in a composition series is therefore independent of the chosen series, and so the composition length, r1 of the group G is well-defined.

• 

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

• 

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

Examples

withGroupTheory:

GAlt4

GA4

(1)

CompositionSeriesG

A41,23,4,1,42,31,23,4

(2)

CompositionLengthG

3

(3)

IsSimplePSL3,3

true

(4)

CompositionSeriesPSL3,3

PSL3,3

(5)

CompositionLengthPSL3,3

1

(6)

csCompositionSeriesDihedralGroup8

csD81,7,5,32,8,6,4,1,2,3,4,5,6,7,81,52,63,74,8

(7)

typecs,SubnormalSeries

true

(8)

typecs,NormalSeries

false

(9)

seqGroupOrderH,H=cs

16,8,4,2,1

(10)

GGroupPerm1,2,3,Perm1,2,Perm4,5,6,Perm4,5,Perm7,8,9,Perm1,4,7,2,5,8,3,6,9,Perm1,4,2,5,3,6

G < a permutation group on 9 letters with 7 generators >

(11)

csCompositionSeriesG

cs < a permutation group on 9 letters with 7 generators > < a permutation group on 9 letters with 7 generators > 1&comma;2&comma;3

(12)

CompositionLengthG

8

(13)

seqGroupOrderH&comma;H=cs

1296,648,324,108,54,27,9,3,1

(14)

Compatibility

• 

The GroupTheory[CompositionSeries] and GroupTheory[CompositionLength] commands were introduced in Maple 2019.

• 

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

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[DihedralGroup]

GroupTheory[IsSimple]

GroupTheory[PSL]

GroupTheory[Series]