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GroupTheory

  

Subgroup

  

create a subgroup

  

TrivialSubgroup

  

create the trivial subgroup of a group

  

Supergroup

  

retrieve a stored subgroup

  

Embedding

  

retrieve the embedding of a group into its supergroup

  

SubgroupMembership

  

determine if an element is embedded into the supergroup

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Subgroup(generators, group)

Supergroup(group)

Embedding(group)

SubgroupMembership(element, group)

TrivialSubgroup( group )

Parameters

generators

-

list or set of generators for the subgroup

group

-

group data structure

element

-

element of the supergroup

Description

• 

The procedure Subgroup creates the subgroup of group generated by generators.

  

Subgroups can also be created by the constructors described on the GroupTheory[Group] help page, by using the supergroup option.

• 

The procedure Supergroup retrieves the supergroup specified upon construction of the group group. If no supergroup was specified, then Supergroup returns undefined.

• 

The procedure Embedding retrieves the embedding of elements of a subgroup into a supergroup. It returns a procedure that can be applied to an element of group to obtain the corresponding element of Supergroupgroup. If Supergroupgroup is undefined, then Embedding will also return undefined.

  

The return value of Embedding can also be interrogated about its domain and codomain: if the return value is e, then Domaine and Codomaine will return the domain and codomain (as groups).

  

The procedure TrivialSubgroup creates the trivial subgroup of the group passed as its argument.

  

For many types of subgroups, Embedding will return the identity mapping.

• 

The procedure SubgroupMembership determines whether a given element of the supergroup of group is an image of Embeddinggroup. This is particularly useful if the embedding is not the identity map - for example, for nontrivial Cayley table subgroups.

  

For types of groups where the meaning of a group element is well-defined independent of the group it lives in, such as permutation groups, SubgroupMembership will just determine group membership without regard for the subgroup structure. For other groups, such as Cayley table groups, if Supergroupgroup is undefined, SubgroupMembership will return undefined.

Examples

withGroupTheory:

g1PermutationGroup1,2,1,2,3,4,5

g11,2,1,2,34,5

(1)

g2Subgroup1,3,2,4,5,g1

g21,3,24,5

(2)

Supergroupg2

1,2,1,2,34,5

(3)

evalbSupergroupg2=evalg1

true

(4)

A nontrivial subgroup of the Klein 4 group as a Cayley table.

g3Group1|2|3|4,2|1|4|3,3|4|1|2,4|3|2|1

g3 < a Cayley table group with 4 elements >

(5)

g4Subgroup3&comma;g3

g4 < a Cayley table group with 1 generator >

(6)

Elementsg3&comma;Elementsg4

1&comma;2&comma;3&comma;4&comma;1&comma;2

(7)

g4 is the subgroup of g3 generated by element 3, but Cayley table groups always have a set of elements 1,...,n. In particular, the elements of the subgroup have been renumbered. We can obtain the original numbering by applying e, defined below:

eEmbeddingg4

e<a group morphism>

(8)

mape&comma;Elementsg4

1&comma;3

(9)

evalbDomaine=g4

true

(10)

evalbCodomaine=g3

true

(11)

In order to test whether an element z of g3 is a member of g4, we cannot just test whether z occurs in Elementsg4 because of this. We can test whether z occurs in the set mape&comma;Elementsg4 constructed above, or (more efficiently) we can use the SubgroupMembership command.

SubgroupMembership3&comma;g4

true

(12)

SubgroupMembership2&comma;g4

false

(13)

Compatibility

• 

The GroupTheory[Subgroup], GroupTheory[Supergroup], GroupTheory[Embedding] and GroupTheory[SubgroupMembership] commands were introduced in Maple 17.

• 

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

• 

The GroupTheory[TrivialSubgroup] command was introduced in Maple 2019.

• 

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

See Also

GroupTheory

GroupTheory[Group]