The normal closure of a subset S in 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 ).

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

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

$H≔⟨\left(1\,3\,2\right)⟩$

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

$3$

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

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

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

$12$

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

$G≔{\mathbf{S}}_3$

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

$N≔⟨\left(2\,3\right)\,\left(1\,2\right)⟩$

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

$6$

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

$3$

$G≔\mathrm{Alt}\left(8\right)$

$G≔{\mathbf{A}}_8$

$$\begin{gathered} \mathbf{do} \\ g≔\mathrm{RandomElement}\left(G\right) \\ \mathbf{until}g\ne \mathrm{Perm}\left(\left[\right]\right) \end{gathered}$$

$\left(1\,3\,5\right)\left(2\,4\,6\,7\,8\right)$

$\mathrm{IsSubgroup}\left(G\,\mathrm{NormalClosure}\left(\left[g\right]\,G\right)\right)$

$\mathrm{true}$

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

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