NormalClosure - Maple Help
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GroupTheory

 NormalClosure
 construct the normal closure of a subgroup or subset of a group

 Calling Sequence NormalClosure( S, G ) NormalClosure( S )

Parameters

 S - a subgroup of G or a set of elements of G G - a permutation group or a Cayley table group

Description

 • 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 ).

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $H≔\mathrm{SylowSubgroup}\left(3,G\right)$
 ${H}{≔}⟨\left({1}{,}{3}{,}{2}\right)⟩$ (2)
 > $\mathrm{GroupOrder}\left(H\right)$
 ${3}$ (3)
 > $N≔\mathrm{NormalClosure}\left(H\right)$
 ${N}{≔}⟨\left({1}{,}{4}{,}{3}\right){,}\left({1}{,}{3}{,}{2}\right)⟩$ (4)
 > $\mathrm{GroupOrder}\left(N\right)$
 ${12}$ (5)
 > $G≔\mathrm{SymmetricGroup}\left(3\right)$
 ${G}{≔}{{\mathbf{S}}}_{{3}}$ (6)
 > $N≔\mathrm{NormalClosure}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right)\right\},G\right)$
 ${N}{≔}⟨\left({2}{,}{3}\right){,}\left({1}{,}{2}\right)⟩$ (7)
 > $\mathrm{GroupOrder}\left(N\right)$
 ${6}$ (8)
 > $\mathrm{GroupOrder}\left(\mathrm{NormalClosure}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2,3\right]\right]\right)\right\},G\right)\right)$
 ${3}$ (9)

The alternating group ${\mathbf{A}}_{8}$ is simple, so the normal closure of any subset with a non-trivial element is the entire group.

 > $G≔\mathrm{Alt}\left(8\right)$
 ${G}{≔}{{\mathbf{A}}}_{{8}}$ (10)
 > $\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}g≔\mathrm{RandomElement}\left(G\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{until}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}g\ne \mathrm{Perm}\left(\left[\right]\right)$
 $\left({1}{,}{4}\right)\left({2}{,}{5}\right)\left({3}{,}{8}{,}{7}\right)$ (11)
 > $\mathrm{IsSubgroup}\left(G,\mathrm{NormalClosure}\left(\left[g\right],G\right)\right)$
 ${\mathrm{true}}$ (12)

Compatibility

 • The GroupTheory[NormalClosure] command was introduced in Maple 17.
 • For more information on Maple 17 changes, see Updates in Maple 17.