A group $G$ is Dedekind if every subgroup of $G$ is normal in $G$. Every Abelian group is obviously a Dedekind group, but non-Abelian Dedekind groups exist.

A group $G$ is Hamiltonian if it is a non-commutative Dedekind group.

The IsDedekind( G ) command attempts to determine whether the group G is Dedekind.  It returns true if G is Dedekind and returns false otherwise.

The IsHamiltonian( G ) command attempts to determine whether the group G is Hamiltonian, returning true if G is Hamiltonian, and false otherwise.

The smallest Hamiltonian group is the quaternion group of order $8$.

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

$\mathrm{IsHamiltonian}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{andmap}\left(\mathrm{IsNormal}\,\mathrm{convert}\left(\mathrm{SubgroupLattice}\left(\mathrm{QuaternionGroup}\left(\right)\right)\,'\mathrm{list}'\right)\,\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{DrawSubgroupLattice}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{IsDedekind}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{IsHamiltonian}\left(\mathrm{CyclicGroup}\left(10\right)\right)$

$\mathrm{false}$

$\mathrm{IsDedekind}\left(\mathrm{CyclicGroup}\left(10\right)\right)$

$\mathrm{true}$

$\mathrm{IsDedekind}\left(\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsHamiltonian}\left(\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{andmap}\left(\mathrm{IsNormal}\,\mathrm{convert}\left(\mathrm{SubgroupLattice}\left(\mathrm{DihedralGroup}\left(4\right)\right)\,'\mathrm{list}'\right)\,\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{DrawSubgroupLattice}\left(\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{IsHamiltonian}\left(\mathrm{SmallGroup}\left(256\,56084\right)\right)$

$\mathrm{true}$

$\mathrm{IsDedekind}\left(\mathrm{SmallGroup}\left(256\,56085\right)\right)$

$\mathrm{false}$

The GroupTheory[IsHamiltonian] and GroupTheory[IsDedekind] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.