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GroupTheory

  

DirectFactors

  

compute the directly indecomposable direct factors of a finite group

  

IsDirectlyIndecomposable

  

determine if a finite group is directly indecomposable

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

DirectFactors( G )

IsDirectlyIndecomposable( G )

Parameters

G

-

a finite group

Description

• 

The DirectFactors( G ) command computes an expression sequence of subgroups of the finite permutation group G such that G is the (internal) direct product of these subgroups, and each subgroup is directly indecomposable.

• 

The Remak-Krull-Schmidt Theorem guarantees that the decomposition is unique up to isomorphism and ordering of the direct factors.

• 

A group is indecomposable if it has no proper non-trivial direct factor. The IsDirectlyIndecomposable( G ) command returns true if G is indecomposable, and false otherwise.

• 

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

Examples

withGroupTheory:

GAlt4

GA4

(1)

DirectFactorsG

A4

(2)

IsDirectlyIndecomposableAlt4

true

(3)

dfDirectFactorsGL2,4

df1,2,34,8,125,10,156,11,137,9,14,1,7,8,11,52,9,12,13,103,14,4,6,15,1,15,4,7,132,5,8,9,63,10,12,14,11

(4)

AreIsomorphicGL2,4,DirectProductdf

true

(5)

IsDirectlyIndecomposabledf1

true

(6)

IsDirectlyIndecomposabledf2

true

(7)

IsDirectlyIndecomposableGL2,4

false

(8)

dfDirectFactorsCyclicGroup24

df1,4,7,10,13,16,19,222,5,8,11,14,17,20,233,6,9,12,15,18,21,24,1,9,172,10,183,11,194,12,205,13,216,14,227,15,238,16,24

(9)

AreIsomorphicCyclicGroup24,DirectProductopdf

true

(10)

IsDirectlyIndecomposableCyclicGroup27

true

(11)

seqIsDirectlyIndecomposableDihedralGroupn,n=2..10

false,true,true,true,false,true,true,true,false

(12)

Compatibility

• 

The GroupTheory[DirectFactors] and GroupTheory[IsDirectlyIndecomposable] commands were introduced in Maple 2019.

• 

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

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[AreIsomorphic]

GroupTheory[CyclicGroup]

GroupTheory[DihedralGroup]

GroupTheory[DirectProduct]