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"
 • outopt : option of the form output = "list" or output = "iterator"

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".
 • 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.

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)
 > $G≔\mathrm{HamiltonianGroup}\left(432,2\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}{,}{4}\right)\left({5}{,}{6}{,}{8}{,}{7}\right){,}\left({1}{,}{5}{,}{3}{,}{8}\right)\left({2}{,}{7}{,}{4}{,}{6}\right){,}\left({9}{,}{10}\right){,}\left({11}{,}{12}{,}{13}\right){,}\left({14}{,}{15}{,}{16}{,}{17}{,}{18}{,}{19}{,}{20}{,}{21}{,}{22}\right)⟩$ (4)
 > $\mathrm{IsHamiltonian}\left(G\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{AllHamiltonianGroups}\left(432,'\mathrm{form}'="fpgroup"\right)$
 $⟨{}{\mathrm{_x3}}{,}{\mathrm{_x4}}{,}{\mathrm{_x5}}{,}{\mathrm{_x6}}{,}{\mathrm{_x7}}{,}{\mathrm{_x8}}{}{\mid }{}{{\mathrm{_x5}}}^{{2}}{,}{{\mathrm{_x6}}}^{{3}}{,}{{\mathrm{_x7}}}^{{3}}{,}{{\mathrm{_x8}}}^{{3}}{,}{{\mathrm{_x3}}}^{{4}}{,}{{\mathrm{_x3}}}^{{2}}{}{{\mathrm{_x4}}}^{{2}}{,}{\mathrm{_x3}}{}{\mathrm{_x4}}{}{{\mathrm{_x3}}}^{{-1}}{}{\mathrm{_x4}}{,}{{\mathrm{_x5}}}^{{-1}}{}{{\mathrm{_x3}}}^{{-1}}{}{\mathrm{_x5}}{}{\mathrm{_x3}}{,}{{\mathrm{_x5}}}^{{-1}}{}{{\mathrm{_x4}}}^{{-1}}{}{\mathrm{_x5}}{}{\mathrm{_x4}}{,}{{\mathrm{_x6}}}^{{-1}}{}{{\mathrm{_x3}}}^{{-1}}{}{\mathrm{_x6}}{}{\mathrm{_x3}}{,}{{\mathrm{_x6}}}^{{-1}}{}{{\mathrm{_x4}}}^{{-1}}{}{\mathrm{_x6}}{}{\mathrm{_x4}}{,}{{\mathrm{_x6}}}^{{-1}}{}{{\mathrm{_x5}}}^{{-1}}{}{\mathrm{_x6}}{}{\mathrm{_x5}}{,}{{\mathrm{_x7}}}^{{-1}}{}{{\mathrm{_x3}}}^{{-1}}{}{\mathrm{_x7}}{}{\mathrm{_x3}}{,}{{\mathrm{_x7}}}^{{-1}}{}{{\mathrm{_x4}}}^{{-1}}{}{\mathrm{_x7}}{}{\mathrm{_x4}}{,}{{\mathrm{_x7}}}^{{-1}}{}{{\mathrm{_x5}}}^{{-1}}{}{\mathrm{_x7}}{}{\mathrm{_x5}}{,}{{\mathrm{_x7}}}^{{-1}}{}{{\mathrm{_x6}}}^{{-1}}{}{\mathrm{_x7}}{}{\mathrm{_x6}}{,}{{\mathrm{_x8}}}^{{-1}}{}{{\mathrm{_x3}}}^{{-1}}{}{\mathrm{_x8}}{}{\mathrm{_x3}}{,}{{\mathrm{_x8}}}^{{-1}}{}{{\mathrm{_x4}}}^{{-1}}{}{\mathrm{_x8}}{}{\mathrm{_x4}}{,}{{\mathrm{_x8}}}^{{-1}}{}{{\mathrm{_x5}}}^{{-1}}{}{\mathrm{_x8}}{}{\mathrm{_x5}}{,}{{\mathrm{_x8}}}^{{-1}}{}{{\mathrm{_x6}}}^{{-1}}{}{\mathrm{_x8}}{}{\mathrm{_x6}}{,}{{\mathrm{_x8}}}^{{-1}}{}{{\mathrm{_x7}}}^{{-1}}{}{\mathrm{_x8}}{}{\mathrm{_x7}}{}⟩{,}⟨{}{\mathrm{_x12}}{,}{\mathrm{_x13}}{,}{\mathrm{_x14}}{,}{\mathrm{_x15}}{,}{\mathrm{_x16}}{}{\mid }{}{{\mathrm{_x14}}}^{{2}}{,}{{\mathrm{_x15}}}^{{3}}{,}{{\mathrm{_x12}}}^{{4}}{,}{{\mathrm{_x12}}}^{{2}}{}{{\mathrm{_x13}}}^{{2}}{,}{\mathrm{_x12}}{}{\mathrm{_x13}}{}{{\mathrm{_x12}}}^{{-1}}{}{\mathrm{_x13}}{,}{{\mathrm{_x14}}}^{{-1}}{}{{\mathrm{_x12}}}^{{-1}}{}{\mathrm{_x14}}{}{\mathrm{_x12}}{,}{{\mathrm{_x14}}}^{{-1}}{}{{\mathrm{_x13}}}^{{-1}}{}{\mathrm{_x14}}{}{\mathrm{_x13}}{,}{{\mathrm{_x15}}}^{{-1}}{}{{\mathrm{_x12}}}^{{-1}}{}{\mathrm{_x15}}{}{\mathrm{_x12}}{,}{{\mathrm{_x15}}}^{{-1}}{}{{\mathrm{_x13}}}^{{-1}}{}{\mathrm{_x15}}{}{\mathrm{_x13}}{,}{{\mathrm{_x15}}}^{{-1}}{}{{\mathrm{_x14}}}^{{-1}}{}{\mathrm{_x15}}{}{\mathrm{_x14}}{,}{{\mathrm{_x16}}}^{{-1}}{}{{\mathrm{_x12}}}^{{-1}}{}{\mathrm{_x16}}{}{\mathrm{_x12}}{,}{{\mathrm{_x16}}}^{{-1}}{}{{\mathrm{_x13}}}^{{-1}}{}{\mathrm{_x16}}{}{\mathrm{_x13}}{,}{{\mathrm{_x16}}}^{{-1}}{}{{\mathrm{_x14}}}^{{-1}}{}{\mathrm{_x16}}{}{\mathrm{_x14}}{,}{{\mathrm{_x16}}}^{{-1}}{}{{\mathrm{_x15}}}^{{-1}}{}{\mathrm{_x16}}{}{\mathrm{_x15}}{,}{{\mathrm{_x16}}}^{{9}}{}⟩{,}⟨{}{\mathrm{_x20}}{,}{\mathrm{_x21}}{,}{\mathrm{_x22}}{,}{\mathrm{_x23}}{}{\mid }{}{{\mathrm{_x22}}}^{{2}}{,}{{\mathrm{_x20}}}^{{4}}{,}{{\mathrm{_x20}}}^{{2}}{}{{\mathrm{_x21}}}^{{2}}{,}{\mathrm{_x20}}{}{\mathrm{_x21}}{}{{\mathrm{_x20}}}^{{-1}}{}{\mathrm{_x21}}{,}{{\mathrm{_x22}}}^{{-1}}{}{{\mathrm{_x20}}}^{{-1}}{}{\mathrm{_x22}}{}{\mathrm{_x20}}{,}{{\mathrm{_x22}}}^{{-1}}{}{{\mathrm{_x21}}}^{{-1}}{}{\mathrm{_x22}}{}{\mathrm{_x21}}{,}{{\mathrm{_x23}}}^{{-1}}{}{{\mathrm{_x20}}}^{{-1}}{}{\mathrm{_x23}}{}{\mathrm{_x20}}{,}{{\mathrm{_x23}}}^{{-1}}{}{{\mathrm{_x21}}}^{{-1}}{}{\mathrm{_x23}}{}{\mathrm{_x21}}{,}{{\mathrm{_x23}}}^{{-1}}{}{{\mathrm{_x22}}}^{{-1}}{}{\mathrm{_x23}}{}{\mathrm{_x22}}{,}{{\mathrm{_x23}}}^{{27}}{}⟩$ (6)
 > $\mathrm{it}≔\mathrm{AllHamiltonianGroups}\left(194400000,'\mathrm{output}'="iterator"\right)$
 ${\mathrm{it}}{≔}{\mathrm{⟨Hamiltonian Groups Iterator for Order 194400000⟩}}$ (7)
 > $\mathrm{nops}\left(\left[\mathrm{seq}\right]\left(\mathrm{it}\right)\right)$
 ${49}$ (8)

Compatibility

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