GroupTheory
HamiltonianGroup
construct a finite Hamiltonian group
NumHamiltonianGroups
find the number of Hamiltonian groups of a given order
AllHamiltonianGroups
find all Hamiltonian groups of a given order
Calling Sequence
Parameters
Options
Description
Examples
Compatibility
HamiltonianGroup( n, k )
NumHamiltonianGroups( n )
AllHamiltonianGroups( n )
n
-
a positive integer
k
formopt : option of the form form = "permgroup" or form = "fpgroup"
outopt : option of the form output = "list" or output = "iterator"
A group is Hamiltonian if it is non-Abelian, and if every subgroup is normal. Every Hamiltonian group has the quaternion group as a direct factor, so the order of every finite Hamiltonian group is a multiple of .
For a positive integer n, the NumHamiltonianGroups( n ) command returns the number of Hamiltonian groups of order n. (This is if n is not a multiple of .)
The HamiltonianGroup( n, k ) command returns the k-th Hamiltonian group of order n. An exception is raised if n is not a multiple of .
The AllHamiltonianGroups( n ) command returns an expression sequence of all the Hamiltonian groups of order n, where n is a positive integer. Note that NULL is returned if n is not a multiple of .
The HamiltonianGroup and AllHamiltonianGroups commands accept an option of the form form = F, where F may be either of the strings "permgroup" (the default), or "fpgroup".
The AllHamiltonianGroups command accepts an option of the form output = "list" (the default) or output = "iterator". By default, a sequence of the Hamiltonian groups of order n is returned. If you pass the option output = "iterator" to AllHamiltonianGroups, then an iterator object is returned instead.
There is an unique Hamiltonian group of each -power greater than or equal to .
There are no Hamiltonian groups of order .
The GroupTheory[HamiltonianGroup], GroupTheory[NumHamiltonianGroups] and GroupTheory[AllHamiltonianGroups] commands were introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
See Also
GroupTheory[IsHamiltonian]
GroupTheory[NumGroups]
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