A group $H$ is a normal subgroup of a group $G$ if $H$ is a subgroup of $G$, and if it is equal to each of its conjugates: $H=H^g$, for all $g$ in $G$.

The IsNormal( H, G ) command tests whether the group H is a normal subgroup of the group G.  It returns true if H is normal in G, and returns false otherwise.  For some pairs H and G of groups, the value FAIL may be returned if IsNormal cannot determine whether H is a normal subgroup of G.

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

$G≔\mathrm{Symm}\left(4\right)$

$G≔{\mathbf{S}}_4$

$H≔\mathrm{Alt}\left(4\right)$

$H≔{\mathbf{A}}_4$

$\mathrm{IsNormal}\left(H\,G\right)$

$\mathrm{true}$

$\mathrm{IsNormal}\left(\mathrm{Alt}\left(5\right)\,G\right)$

$\mathrm{false}$

$\mathrm{IsSubgroup}\left(\mathrm{Alt}\left(5\right)\,G\right)$

$\mathrm{false}$

$G≔\mathrm{Symm}\left(5\right)$

$G≔{\mathbf{S}}_5$

$H≔\mathrm{DihedralGroup}\left(5\right)$

$H≔{\mathbf{D}}_5$

$\mathrm{IsNormal}\left(H\,G\right)$

$\mathrm{false}$

$\mathrm{IsSubgroup}\left(H\,G\right)$

$\mathrm{true}$

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

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