The normal closure of a subset S of a group G is the smallest normal subgroup of G containing S.

The NormalClosure( G ) command constructs the normal closure of S in G.

The group G must be an instance of a permutation group or a Cayley table group.

If S is a subgroup of a group, then the one-argument form NormalClosure( S ) constructs the normal closure of S in the parent group Supergroup( S ).

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$H≔\mathrm{SylowSubgroup}\left(3\,G\right)$

$H≔\mathrm{<a\; permutation\; group\; on\; 4\; letters>}$

$\mathrm{GroupOrder}\left(H\right)$

$3$

$N≔\mathrm{NormalClosure}\left(H\right)$

$N≔⟨\left(1\&comma;3\&comma;2\right)\&comma;\left(1\&comma;4\&comma;3\right)⟩$

$\mathrm{GroupOrder}\left(N\right)$

$12$

$G≔\mathrm{SymmetricGroup}\left(3\right)$

$G≔{\mathbf{S}}_3$

$N≔\mathrm{NormalClosure}\left(\left\{\mathrm{Perm}\left(\left[\left[1\&comma;2\right]\right]\right)\right\}\&comma;G\right)$

$N≔⟨\left(1\&comma;2\right)\&comma;\left(2\&comma;3\right)⟩$

$\mathrm{GroupOrder}\left(N\right)$

$6$

$\mathrm{GroupOrder}\left(\mathrm{NormalClosure}\left(\left\{\mathrm{Perm}\left(\left[\left[1\&comma;2\&comma;3\right]\right]\right)\right\}\&comma;G\right)\right)$

$3$

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

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