Normaliser - Maple Help

GroupTheory

 Normaliser
 construct the normaliser of a subgroup of a group

 Calling Sequence Normaliser( H, G ) NormaliserSubgroup( H, G ) NormalizerSubgroup( H, G )

Parameters

 G - a permutation group or a Cayley table group H - a permutation group or a Cayley table group

Description

 • The normaliser of a subgroup $H$ of $G$ is the set of elements $g\in G$ for which commutation by $g$ induces an automorphism on $H$. That is, $\frac{1}{g}·H·g=H$, or equivalently, $H·g=g·H$, or equivalently, for all $h\in H$ we have $\frac{1}{g}·h·g\in H$.
 • The Normaliser( H, G ) command constructs the normaliser of H in G. The group G must be a group given by a Cayley table or a permutation group.
 • The NormaliserSubgroup and NormalizerSubgroup commands are provided as aliases. Note that Normalizer is a different command, unrelated to the $\mathrm{GroupTheory}$ package; because it is an environment variable, the $\mathrm{GroupTheory}$ package cannot provide a command with this name.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{G1}≔\mathrm{SymmetricGroup}\left(5\right)$
 ${\mathrm{G1}}{≔}{{\mathbf{S}}}_{{5}}$ (1)
 > $\mathrm{elements}≔\mathrm{convert}\left(\mathrm{Elements}\left(\mathrm{G1}\right),'\mathrm{list}'\right):$
 > $\mathrm{CT}≔\mathrm{CayleyTable}\left(\mathrm{G1},'\mathrm{elements}'=\mathrm{elements}\right):$
 > $\mathrm{G2}≔\mathrm{Group}\left(\mathrm{CT}\right)$
 ${\mathrm{G2}}{≔}{\mathrm{< a Cayley table group with 120 elements >}}$ (2)

Now the elements of $\mathrm{G2}$ correspond to the list $\mathrm{elements}$ in the given order. We can find the elements corresponding to the permutations $\left[\left[1,2\right]\right]$ and $\left[\left[1,3\right]\right]$ by looking up their positions in $\mathrm{elements}$, in order to construct the symmetric group on 3 letters as a subgroup $H$.

 > $\mathrm{generators}≔\mathrm{map}\left(\mathrm{ListTools}:-\mathrm{Search},\left[\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,3\right]\right]\right)\right],\mathrm{elements}\right)$
 ${\mathrm{generators}}{≔}\left[{120}{,}{86}\right]$ (3)
 > $H≔\mathrm{Subgroup}\left(\mathrm{generators},\mathrm{G2}\right)$
 ${H}{≔}{\mathrm{< a Cayley table group with 2 generators >}}$ (4)
 > $N≔\mathrm{Normaliser}\left(H,\mathrm{G2}\right)$
 ${N}{≔}{{N}}_{{\mathrm{< a Cayley table group with 120 elements >}}}{}\left({\mathrm{< a Cayley table group with 2 generators >}}\right)$ (5)

Since $N$ is itself a Cayley table group, it is most useful to inspect the images of the elements under the Embedding.

 > $\mathrm{elementsN}≔\mathrm{map}\left(\mathrm{Embedding}\left(N\right),\mathrm{Elements}\left(N\right)\right)$
 ${\mathrm{elementsN}}{≔}\left\{{45}{,}{80}{,}{81}{,}{86}{,}{100}{,}{105}{,}{106}{,}{108}{,}{109}{,}{114}{,}{115}{,}{120}\right\}$ (6)
 > $\mathrm{elements}\left[\mathrm{convert}\left(\mathrm{elementsN},'\mathrm{list}'\right)\right]$
 $\left[\left({1}{,}{3}\right)\left({4}{,}{5}\right){,}\left({1}{,}{3}{,}{2}\right){,}\left({1}{,}{3}{,}{2}\right)\left({4}{,}{5}\right){,}\left({1}{,}{3}\right){,}\left({1}{,}{2}\right)\left({4}{,}{5}\right){,}\left({1}{,}{2}{,}{3}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right){,}\left(\right){,}\left({4}{,}{5}\right){,}\left({2}{,}{3}\right){,}\left({2}{,}{3}\right)\left({4}{,}{5}\right){,}\left({1}{,}{2}\right)\right]$ (7)

$N$ is the direct product of $H$ and the 2-element subgroup generated by the transposition $\left[\left[4,5\right]\right]$.

 > $\mathrm{IsAbelian}\left(N\right)$
 ${\mathrm{false}}$ (8)
 > $\mathrm{GroupOrder}\left(N\right)$
 ${12}$ (9)

Compatibility

 • The GroupTheory[Normaliser] command was introduced in Maple 17.