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 $\mathrm{group}$ to obtain the corresponding element of $\mathrm{Supergroup}\left(\mathrm{group}\right)$. If $\mathrm{Supergroup}\left(\mathrm{group}\right)$ 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 $\mathrm{Domain}\left(e\right)$ and $\mathrm{Codomain}\left(e\right)$ 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 $\mathrm{group}$ is an image of $\mathrm{Embedding}\left(\mathrm{group}\right)$. 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 $\mathrm{Supergroup}\left(\mathrm{group}\right)$ is undefined, SubgroupMembership will return undefined.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{g1}≔\mathrm{PermutationGroup}\left(\left\{\left[\left[1\,2\right]\right]\,\left[\left[1\,2\,3\right]\,\left[4\,5\right]\right]\right\}\right)$

$\mathrm{g1}≔⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩$

$\mathrm{g2}≔\mathrm{Subgroup}\left(\left\{\left[\left[1\,3\,2\right]\,\left[4\,5\right]\right]\right\}\,\mathrm{g1}\right)$

$\mathrm{g2}≔⟨\left(1\,3\,2\right)\left(4\,5\right)⟩$

$\mathrm{Supergroup}\left(\mathrm{g2}\right)$

$⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩$

$\mathrm{evalb}\left(\mathrm{Supergroup}\left(\mathrm{g2}\right)=\mathrm{eval}\left(\mathrm{g1}\right)\right)$

$\mathrm{true}$

$\mathrm{g3}≔\mathrm{Group}\left(⟨⟨1\left|2\right|3|4⟩\,⟨2\left|1\right|4|3⟩\,⟨3\left|4\right|1|2⟩\,⟨4\left|3\right|2|1⟩⟩\right)$

$\mathrm{g3}≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 4\; elements\; >}$

$\mathrm{g4}≔\mathrm{Subgroup}\left(\left[3\right]\&comma;\mathrm{g3}\right)$

$\mathrm{g4}≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 1\; generator\; >}$

$\left[\mathrm{Elements}\left(\mathrm{g3}\right)\&comma;\mathrm{Elements}\left(\mathrm{g4}\right)\right]$

$\left[\left\{1\&comma;2\&comma;3\&comma;4\right\}\&comma;\left\{1\&comma;2\right\}\right]$

$e≔\mathrm{Embedding}\left(\mathrm{g4}\right)$

$e≔\mathrm{<a\; group\; morphism>}$

$\mathrm{map}\left(e\&comma;\mathrm{Elements}\left(\mathrm{g4}\right)\right)$

$\left\{1\&comma;3\right\}$

$\mathrm{evalb}\left(\mathrm{Domain}\left(e\right)=\mathrm{g4}\right)$

$\mathrm{true}$

$\mathrm{evalb}\left(\mathrm{Codomain}\left(e\right)=\mathrm{g3}\right)$

$\mathrm{true}$

$\mathrm{SubgroupMembership}\left(3\&comma;\mathrm{g4}\right)$

$\mathrm{true}$

$\mathrm{SubgroupMembership}\left(2\&comma;\mathrm{g4}\right)$

$\mathrm{false}$

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.