construct the normaliser of a subgroup of a group
Normaliser( H, G )
NormaliserSubgroup( H, G )
NormalizerSubgroup( H, G )
a permutation group or a Cayley table group
The normaliser of a subgroup H of G is the set of elements g∈G 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 h∈H we have 1g·h·g∈H.
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.
G1 ≔ SymmetricGroup⁡5
elements ≔ convert⁡Elements⁡G1,'list':
CT ≔ CayleyTable⁡G1,'elements'=elements:
G2 ≔ Group⁡CT
G2≔ < a Cayley table group with 120 elements >
Now the elements of G2 correspond to the list elements in the given order. We can find the elements corresponding to the permutations 1,2 and 1,3 by looking up their positions in elements, in order to construct the symmetric group on 3 letters as a subgroup H.
generators ≔ map⁡ListTools:-Search,Perm⁡1,2,Perm⁡1,3,elements
H ≔ Subgroup⁡generators,G2
H≔ < a Cayley table group with 2 generators >
N ≔ Normaliser⁡H,G2
N≔N < a Cayley table group with 120 elements > ⁡ < a Cayley table group with 2 generators >
Since N is itself a Cayley table group, it is most useful to inspect the images of the elements under the Embedding.
elementsN ≔ map⁡Embedding⁡N,Elements⁡N
N is the direct product of H and the 2-element subgroup generated by the transposition 4,5.
The GroupTheory[Normaliser] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
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