IsDedekind - Maple Help

GroupTheory

 IsHamiltonian
 attempt to determine whether a group is Hamiltonian
 IsDedekind
 attempt to determine whether a group is Dedekind

 Calling Sequence IsDedekind( G ) IsHamiltonian( G )

Parameters

 G - a permutation group

Description

 • 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 $8$.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{IsHamiltonian}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{andmap}\left(\mathrm{IsNormal},\mathrm{convert}\left(\mathrm{SubgroupLattice}\left(\mathrm{QuaternionGroup}\left(\right)\right),'\mathrm{list}'\right),\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (2)

The fact that this group is Hamiltonian is visible from the subgroup lattice:

 > $\mathrm{DrawSubgroupLattice}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 > $\mathrm{IsDedekind}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsHamiltonian}\left(\mathrm{CyclicGroup}\left(10\right)\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{IsDedekind}\left(\mathrm{CyclicGroup}\left(10\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsDedekind}\left(\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{IsHamiltonian}\left(\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{false}}$ (7)
 > $\mathrm{andmap}\left(\mathrm{IsNormal},\mathrm{convert}\left(\mathrm{SubgroupLattice}\left(\mathrm{DihedralGroup}\left(4\right)\right),'\mathrm{list}'\right),\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{false}}$ (8)

You can see that the dihedral group of order $8$ is not Hamiltonian by looking at its subgroup lattice.

 > $\mathrm{DrawSubgroupLattice}\left(\mathrm{DihedralGroup}\left(4\right)\right)$
 > $\mathrm{IsHamiltonian}\left(\mathrm{SmallGroup}\left(256,56084\right)\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{IsDedekind}\left(\mathrm{SmallGroup}\left(256,56085\right)\right)$
 ${\mathrm{false}}$ (10)
 > 

Compatibility

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