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.

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

$\mathrm{G1}≔\mathrm{SymmetricGroup}\left(5\right)$

$\mathrm{G1}≔{\mathbf{S}}_5$

$\mathrm{elements}≔\mathrm{convert}\left(\mathrm{Elements}\left(\mathrm{G1}\right)\,'\mathrm{list}'\right)\:$

$\mathrm{CT}≔\mathrm{CayleyTable}\left(\mathrm{G1}\,'\mathrm{elements}'=\mathrm{elements}\right)\:$

$\mathrm{G2}≔\mathrm{Group}\left(\mathrm{CT}\right)$

$\mathrm{G2}≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 120\; elements\; >}$

$\mathrm{generators}≔\mathrm{map}\left(\mathrm{ListTools}:-\mathrm{Search}\&comma;\left[\mathrm{Perm}\left(\left[\left[1\&comma;2\right]\right]\right)\&comma;\mathrm{Perm}\left(\left[\left[1\&comma;3\right]\right]\right)\right]\&comma;\mathrm{elements}\right)$

$\mathrm{generators}≔\left[119\&comma;107\right]$

$H≔\mathrm{Subgroup}\left(\mathrm{generators}\&comma;\mathrm{G2}\right)$

$H≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 2\; generators\; >}$

$N≔\mathrm{Normaliser}\left(H\&comma;\mathrm{G2}\right)$

$N≔N_{\mathrm{<\; a\; Cayley\; table\; group\; with\; 120\; elements\; >}}\left(\mathrm{<\; a\; Cayley\; table\; group\; with\; 2\; generators\; >}\right)$

$\mathrm{elementsN}≔\mathrm{map}\left(\mathrm{Embedding}\left(N\right)\&comma;\mathrm{Elements}\left(N\right)\right)$

$\mathrm{elementsN}≔\left\{15\&comma;16\&comma;26\&comma;28\&comma;54\&comma;56\&comma;81\&comma;84\&comma;87\&comma;106\&comma;107\&comma;119\right\}$

$\mathrm{elements}\left[\mathrm{convert}\left(\mathrm{elementsN}\&comma;'\mathrm{list}'\right)\right]$

$\left[\left(4\&comma;5\right)\&comma;\left(\right)\&comma;\left(1\&comma;2\&comma;3\right)\&comma;\left(1\&comma;2\&comma;3\right)\left(4\&comma;5\right)\&comma;\left(2\&comma;3\right)\&comma;\left(2\&comma;3\right)\left(4\&comma;5\right)\&comma;\left(1\&comma;3\&comma;2\right)\left(4\&comma;5\right)\&comma;\left(1\&comma;3\&comma;2\right)\&comma;\left(1\&comma;3\right)\left(4\&comma;5\right)\&comma;\left(1\&comma;2\right)\left(4\&comma;5\right)\&comma;\left(1\&comma;3\right)\&comma;\left(1\&comma;2\right)\right]$

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

$\mathrm{false}$

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

$12$

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

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