AllHamiltonianGroups - Maple Help

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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 HamiltonianGroup( n, k ) NumHamiltonianGroups( n ) AllHamiltonianGroups( n )

Parameters

 n - a positive integer k - a positive integer

Options

 • formopt : option of the form form = "permgroup" or form = "fpgroup"

Description

 • 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 $8$.
 • For a positive integer n, the NumHamiltonianGroups( n ) command returns the number of Hamiltonian groups of order n. (This is $0$ if n is not a multiple of $8$.)
 • 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 $8$.
 • 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 $8$.
 • 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".

Examples

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

There is an unique Hamiltonian group of each $2$-power greater than or equal to $8$.

 > $\mathrm{seq}\left(\mathrm{NumHamiltonianGroups}\left({2}^{i}\right),i=1..20\right)$
 ${0}{,}{0}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}{,}{1}$ (1)

There are no Hamiltonian groups of order $25$.

 > $\mathrm{NumHamiltonianGroups}\left(25\right)$
 ${0}$ (2)
 > $\mathrm{NumHamiltonianGroups}\left(432\right)$
 ${3}$ (3)
 > $G≔\mathrm{HamiltonianGroup}\left(432,2\right)$
 ${G}{≔}{\mathrm{< a permutation group on 22 letters with 5 generators >}}$ (4)
 > $\mathrm{IsHamiltonian}\left(G\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{AllHamiltonianGroups}\left(432,'\mathrm{form}'="fpgroup"\right)$
 $⟨{}{i}{,}{j}{,}{\mathrm{_g}}{,}{\mathrm{_a1}}{,}{\mathrm{_a2}}{,}{\mathrm{_a3}}{}{\mid }{}{{\mathrm{_g}}}^{{2}}{,}{{\mathrm{_a1}}}^{{3}}{,}{{\mathrm{_a2}}}^{{3}}{,}{{\mathrm{_a3}}}^{{3}}{,}{{i}}^{{4}}{,}{{i}}^{{2}}{}{{j}}^{{2}}{,}{i}{}{j}{}{{i}}^{{-1}}{}{j}{,}{{\mathrm{_a1}}}^{{-1}}{}{{\mathrm{_g}}}^{{-1}}{}{\mathrm{_a1}}{}{\mathrm{_g}}{,}{{\mathrm{_a1}}}^{{-1}}{}{{i}}^{{-1}}{}{\mathrm{_a1}}{}{i}{,}{{\mathrm{_a1}}}^{{-1}}{}{{j}}^{{-1}}{}{\mathrm{_a1}}{}{j}{,}{{\mathrm{_a2}}}^{{-1}}{}{{\mathrm{_a1}}}^{{-1}}{}{\mathrm{_a2}}{}{\mathrm{_a1}}{,}{{\mathrm{_a2}}}^{{-1}}{}{{\mathrm{_g}}}^{{-1}}{}{\mathrm{_a2}}{}{\mathrm{_g}}{,}{{\mathrm{_a2}}}^{{-1}}{}{{i}}^{{-1}}{}{\mathrm{_a2}}{}{i}{,}{{\mathrm{_a2}}}^{{-1}}{}{{j}}^{{-1}}{}{\mathrm{_a2}}{}{j}{,}{{\mathrm{_a3}}}^{{-1}}{}{{\mathrm{_a1}}}^{{-1}}{}{\mathrm{_a3}}{}{\mathrm{_a1}}{,}{{\mathrm{_a3}}}^{{-1}}{}{{\mathrm{_a2}}}^{{-1}}{}{\mathrm{_a3}}{}{\mathrm{_a2}}{,}{{\mathrm{_a3}}}^{{-1}}{}{{\mathrm{_g}}}^{{-1}}{}{\mathrm{_a3}}{}{\mathrm{_g}}{,}{{\mathrm{_a3}}}^{{-1}}{}{{i}}^{{-1}}{}{\mathrm{_a3}}{}{i}{,}{{\mathrm{_a3}}}^{{-1}}{}{{j}}^{{-1}}{}{\mathrm{_a3}}{}{j}{,}{{i}}^{{-1}}{}{{\mathrm{_g}}}^{{-1}}{}{i}{}{\mathrm{_g}}{,}{{j}}^{{-1}}{}{{\mathrm{_g}}}^{{-1}}{}{j}{}{\mathrm{_g}}{}⟩{,}⟨{}{i}{,}{j}{,}{\mathrm{_g0}}{,}{\mathrm{_a1}}{,}{\mathrm{_a2}}{}{\mid }{}{{\mathrm{_g0}}}^{{2}}{,}{{\mathrm{_a1}}}^{{3}}{,}{{i}}^{{4}}{,}{{i}}^{{2}}{}{{j}}^{{2}}{,}{i}{}{j}{}{{i}}^{{-1}}{}{j}{,}{{\mathrm{_a1}}}^{{-1}}{}{{\mathrm{_g0}}}^{{-1}}{}{\mathrm{_a1}}{}{\mathrm{_g0}}{,}{{\mathrm{_a1}}}^{{-1}}{}{{i}}^{{-1}}{}{\mathrm{_a1}}{}{i}{,}{{\mathrm{_a1}}}^{{-1}}{}{{j}}^{{-1}}{}{\mathrm{_a1}}{}{j}{,}{{\mathrm{_a2}}}^{{-1}}{}{{\mathrm{_a1}}}^{{-1}}{}{\mathrm{_a2}}{}{\mathrm{_a1}}{,}{{\mathrm{_a2}}}^{{-1}}{}{{\mathrm{_g0}}}^{{-1}}{}{\mathrm{_a2}}{}{\mathrm{_g0}}{,}{{\mathrm{_a2}}}^{{-1}}{}{{i}}^{{-1}}{}{\mathrm{_a2}}{}{i}{,}{{\mathrm{_a2}}}^{{-1}}{}{{j}}^{{-1}}{}{\mathrm{_a2}}{}{j}{,}{{i}}^{{-1}}{}{{\mathrm{_g0}}}^{{-1}}{}{i}{}{\mathrm{_g0}}{,}{{j}}^{{-1}}{}{{\mathrm{_g0}}}^{{-1}}{}{j}{}{\mathrm{_g0}}{,}{{\mathrm{_a2}}}^{{9}}{}⟩{,}⟨{}{i}{,}{j}{,}{\mathrm{_g1}}{,}{\mathrm{_a1}}{}{\mid }{}{{\mathrm{_g1}}}^{{2}}{,}{{i}}^{{4}}{,}{{i}}^{{2}}{}{{j}}^{{2}}{,}{i}{}{j}{}{{i}}^{{-1}}{}{j}{,}{{\mathrm{_a1}}}^{{-1}}{}{{\mathrm{_g1}}}^{{-1}}{}{\mathrm{_a1}}{}{\mathrm{_g1}}{,}{{\mathrm{_a1}}}^{{-1}}{}{{i}}^{{-1}}{}{\mathrm{_a1}}{}{i}{,}{{\mathrm{_a1}}}^{{-1}}{}{{j}}^{{-1}}{}{\mathrm{_a1}}{}{j}{,}{{i}}^{{-1}}{}{{\mathrm{_g1}}}^{{-1}}{}{i}{}{\mathrm{_g1}}{,}{{j}}^{{-1}}{}{{\mathrm{_g1}}}^{{-1}}{}{j}{}{\mathrm{_g1}}{,}{{\mathrm{_a1}}}^{{27}}{}⟩$ (6)

Compatibility

 • The GroupTheory[HamiltonianGroup], GroupTheory[NumHamiltonianGroups] and GroupTheory[AllHamiltonianGroups] commands were introduced in Maple 2019.