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GroupTheory

  

Normaliser

  

construct the normaliser of a subgroup of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Normaliser( H, G )

NormaliserSubgroup( H, G )

NormalizerSubgroup( H, G )

Parameters

G

-

a permutation group or a Cayley table group

H

-

a permutation group or a Cayley table group

Description

• 

The normaliser of a subgroup H of G is the set of elements gG for which commutation by g induces an automorphism on H. That is, 1g·H·g=H, or equivalently, H·g=g·H, or equivalently, for all hH we have 1g·h·gH.

• 

The Normaliser( H, G ) command constructs the normaliser of H in G. The group G must be a group given by a Cayley table or a permutation group.

• 

The NormaliserSubgroup and NormalizerSubgroup commands are provided as aliases. Note that Normalizer is a different command, unrelated to the GroupTheory package; because it is an environment variable, the GroupTheory package cannot provide a command with this name.

Examples

withGroupTheory:

G1SymmetricGroup5

G1S5

(1)

elementsconvertElementsG1,list:

CTCayleyTableG1,elements=elements:

G2GroupCT

G2 < a Cayley table group with 120 elements >

(2)

Now the elements of G2 correspond to the list elements in the given order. We can find the elements corresponding to the permutations 1&comma;2 and 1&comma;3 by looking up their positions in elements, in order to construct the symmetric group on 3 letters as a subgroup H.

generatorsmapListTools:-Search&comma;Perm1&comma;2&comma;Perm1&comma;3&comma;elements

generators120&comma;117

(3)

HSubgroupgenerators&comma;G2

H < a Cayley table group with 2 generators >

(4)

NNormaliserH&comma;G2

NN < a Cayley table group with 120 elements > < a Cayley table group with 2 generators >

(5)

Since N is itself a Cayley table group, it is most useful to inspect the images of the elements under the Embedding.

elementsNmapEmbeddingN&comma;ElementsN

elementsN1&comma;12&comma;25&comma;28&comma;49&comma;96&comma;101&comma;102&comma;105&comma;116&comma;117&comma;120

(6)

elementsconvertelementsN&comma;list

1&comma;3&comma;24&comma;5&comma;1&comma;34&comma;5&comma;2&comma;34&comma;5&comma;1&comma;2&comma;34&comma;5&comma;1&comma;24&comma;5&comma;2&comma;3&comma;4&comma;5&comma;1&comma;2&comma;3&comma;1&comma;3&comma;2&comma;&comma;1&comma;3&comma;1&comma;2

(7)

N is the direct product of H and the 2-element subgroup generated by the transposition 4&comma;5.

IsAbelianN

false

(8)

GroupOrderN

12

(9)

Compatibility

• 

The GroupTheory[Normaliser] command was introduced in Maple 17.

• 

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

See Also

GroupTheory

GroupTheory[Centralizer]