Named Finitely Presented Groups#

Construct groups of small order and “named” groups as quotients of free groups. These groups are available through tab completion by typing groups.presentation.<tab> or by importing the required methods. Tab completion is made available through Sage’s group catalog. Some examples are engineered from entries in [TW1980].

Groups available as finite presentations:

Alternating group, \(A_n\) of order \(n!/2\) – groups.presentation.Alternating
the \(n\)-fruit Cactus group, a standard notation for which is \(J_n\) – groups.presentation.Cactus
Cyclic group, \(C_n\) of order \(n\) – groups.presentation.Cyclic
Dicyclic group, nonabelian groups of order \(4n\) with a unique element of order 2 – groups.presentation.DiCyclic
Dihedral group, \(D_n\) of order \(2n\) – groups.presentation.Dihedral
Finitely generated abelian group, \(\ZZ_{n_1} \times \ZZ_{n_2} \times \cdots \times \ZZ_{n_k}\) – groups.presentation.FGAbelian
Finitely generated Heisenberg group – groups.presentation.Heisenberg
Klein four group, \(C_2 \times C_2\) – groups.presentation.KleinFour
Quaternion group of order 8 – groups.presentation.Quaternion
Symmetric group, \(S_n\) of order \(n!\) – groups.presentation.Symmetric

AUTHORS:

Davis Shurbert (2013-06-21): initial version

EXAMPLES:

Sage

sage: groups.presentation.Cyclic(4)
Finitely presented group < a | a^4 >

Python

>>> from sage.all import *
>>> groups.presentation.Cyclic(Integer(4))
Finitely presented group < a | a^4 >

You can also import the desired functions:

Sage

sage: from sage.groups.finitely_presented_named import CyclicPresentation
sage: CyclicPresentation(4)
Finitely presented group < a | a^4 >

Python

>>> from sage.all import *
>>> from sage.groups.finitely_presented_named import CyclicPresentation
>>> CyclicPresentation(Integer(4))
Finitely presented group < a | a^4 >

sage.groups.finitely_presented_named.AlternatingPresentation(n)[source]#

Build the Alternating group of order \(n!/2\) as a finitely presented group.

INPUT:

n – the size of the underlying set of arbitrary symbols being acted on by the Alternating group of order \(n!/2\)

OUTPUT:

Alternating group as a finite presentation, implementation uses GAP to find an isomorphism from a permutation representation to a finitely presented group representation. Due to this fact, the exact output presentation may not be the same for every method call on a constant n.

EXAMPLES:

Sage

sage: A6 = groups.presentation.Alternating(6)
sage: A6.as_permutation_group().is_isomorphic(AlternatingGroup(6)), A6.order()
(True, 360)

Python

>>> from sage.all import *
>>> A6 = groups.presentation.Alternating(Integer(6))
>>> A6.as_permutation_group().is_isomorphic(AlternatingGroup(Integer(6))), A6.order()
(True, 360)

sage.groups.finitely_presented_named.BinaryDihedralPresentation(n)[source]#

Build a binary dihedral group of order \(4n\) as a finitely presented group.

The binary dihedral group \(BD_n\) has the following presentation (note that there is a typo in [Sun2010]):

\[BD_n = \langle x, y, z | x^2 = y^2 = z^n = x y z \rangle.\]

INPUT:

n – the value \(n\)

OUTPUT: the binary dihedral group of order \(4n\) as finite presentation

EXAMPLES:

Sage

sage: groups.presentation.BinaryDihedral(9)
Finitely presented group < x, y, z | x^-2*y^2, x^-2*z^9, x^-1*y*z >

Python

>>> from sage.all import *
>>> groups.presentation.BinaryDihedral(Integer(9))
Finitely presented group < x, y, z | x^-2*y^2, x^-2*z^9, x^-1*y*z >

sage.groups.finitely_presented_named.CactusPresentation(n)[source]#

Build the \(n\)-fruit cactus group as a finitely presented group.

OUTPUT: Cactus group \(J_n\) as a finitely presented group

EXAMPLES:

Sage

sage: J3 = groups.presentation.Cactus(3); J3                                    # needs sage.graphs
Finitely presented group < s12, s13, s23 |
 s12^2, s13^2, s23^2, s13*s12*s13^-1*s23^-1, s13*s23*s13^-1*s12^-1 >

Python

>>> from sage.all import *
>>> J3 = groups.presentation.Cactus(Integer(3)); J3                                    # needs sage.graphs
Finitely presented group < s12, s13, s23 |
 s12^2, s13^2, s23^2, s13*s12*s13^-1*s23^-1, s13*s23*s13^-1*s12^-1 >

sage.groups.finitely_presented_named.CyclicPresentation(n)[source]#

Build cyclic group of order \(n\) as a finitely presented group.

INPUT:

n – the order of the cyclic presentation to be returned

OUTPUT: the cyclic group of order \(n\) as finite presentation

EXAMPLES:

Sage

sage: groups.presentation.Cyclic(10)
Finitely presented group < a | a^10 >
sage: n = 8; C = groups.presentation.Cyclic(n)
sage: C.as_permutation_group().is_isomorphic(CyclicPermutationGroup(n))
True

Python

>>> from sage.all import *
>>> groups.presentation.Cyclic(Integer(10))
Finitely presented group < a | a^10 >
>>> n = Integer(8); C = groups.presentation.Cyclic(n)
>>> C.as_permutation_group().is_isomorphic(CyclicPermutationGroup(n))
True

sage.groups.finitely_presented_named.DiCyclicPresentation(n)[source]#

Build the dicyclic group of order \(4n\), for \(n \geq 2\), as a finitely presented group.

INPUT:

n – positive integer, 2 or greater, determining the order of the group (\(4n\))

OUTPUT: the dicyclic group of order \(4n\) is defined by the presentation

\[\langle a, x \mid a^{2n}=1, x^{2}=a^{n}, x^{-1}ax=a^{-1} \rangle\]

Note

This group is also available as a permutation group via groups.permutation.DiCyclic.

EXAMPLES:

Sage

sage: D = groups.presentation.DiCyclic(9); D
Finitely presented group < a, b | a^18, b^2*a^-9, b^-1*a*b*a >
sage: D.as_permutation_group().is_isomorphic(groups.permutation.DiCyclic(9))
True

Python

>>> from sage.all import *
>>> D = groups.presentation.DiCyclic(Integer(9)); D
Finitely presented group < a, b | a^18, b^2*a^-9, b^-1*a*b*a >
>>> D.as_permutation_group().is_isomorphic(groups.permutation.DiCyclic(Integer(9)))
True

sage.groups.finitely_presented_named.DihedralPresentation(n)[source]#

Build the Dihedral group of order \(2n\) as a finitely presented group.

INPUT:

n – the size of the set that \(D_n\) is acting on

OUTPUT: Dihedral group of order \(2n\)

EXAMPLES:

Sage

sage: D = groups.presentation.Dihedral(7); D
Finitely presented group < a, b | a^7, b^2, (a*b)^2 >
sage: D.as_permutation_group().is_isomorphic(DihedralGroup(7))
True

Python

>>> from sage.all import *
>>> D = groups.presentation.Dihedral(Integer(7)); D
Finitely presented group < a, b | a^7, b^2, (a*b)^2 >
>>> D.as_permutation_group().is_isomorphic(DihedralGroup(Integer(7)))
True

sage.groups.finitely_presented_named.FinitelyGeneratedAbelianPresentation(int_list)[source]#

Return canonical presentation of finitely generated abelian group.

INPUT:

int_list – list of integers defining the group to be returned, the defining list is reduced to the invariants of the input list before generating the corresponding group

OUTPUT:

Finitely generated abelian group, \(\ZZ_{n_1} \times \ZZ_{n_2} \times \cdots \times \ZZ_{n_k}\) as a finite presentation, where \(n_i\) forms the invariants of the input list.

EXAMPLES:

Sage

sage: groups.presentation.FGAbelian([2,2])
Finitely presented group < a, b | a^2, b^2, a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([2,3])
Finitely presented group < a | a^6 >
sage: groups.presentation.FGAbelian([2,4])
Finitely presented group < a, b | a^2, b^4, a^-1*b^-1*a*b >

Python

>>> from sage.all import *
>>> groups.presentation.FGAbelian([Integer(2),Integer(2)])
Finitely presented group < a, b | a^2, b^2, a^-1*b^-1*a*b >
>>> groups.presentation.FGAbelian([Integer(2),Integer(3)])
Finitely presented group < a | a^6 >
>>> groups.presentation.FGAbelian([Integer(2),Integer(4)])
Finitely presented group < a, b | a^2, b^4, a^-1*b^-1*a*b >

You can create free abelian groups:

Sage

sage: groups.presentation.FGAbelian([0])
Finitely presented group < a |  >
sage: groups.presentation.FGAbelian([0,0])
Finitely presented group < a, b | a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([0,0,0])
Finitely presented group < a, b, c | a^-1*b^-1*a*b, a^-1*c^-1*a*c, b^-1*c^-1*b*c >

Python

>>> from sage.all import *
>>> groups.presentation.FGAbelian([Integer(0)])
Finitely presented group < a |  >
>>> groups.presentation.FGAbelian([Integer(0),Integer(0)])
Finitely presented group < a, b | a^-1*b^-1*a*b >
>>> groups.presentation.FGAbelian([Integer(0),Integer(0),Integer(0)])
Finitely presented group < a, b, c | a^-1*b^-1*a*b, a^-1*c^-1*a*c, b^-1*c^-1*b*c >

And various infinite abelian groups:

Sage

sage: groups.presentation.FGAbelian([0,2])
Finitely presented group < a, b | a^2, a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([0,2,2])
Finitely presented group < a, b, c | a^2, b^2, a^-1*b^-1*a*b, a^-1*c^-1*a*c, b^-1*c^-1*b*c >

Python

>>> from sage.all import *
>>> groups.presentation.FGAbelian([Integer(0),Integer(2)])
Finitely presented group < a, b | a^2, a^-1*b^-1*a*b >
>>> groups.presentation.FGAbelian([Integer(0),Integer(2),Integer(2)])
Finitely presented group < a, b, c | a^2, b^2, a^-1*b^-1*a*b, a^-1*c^-1*a*c, b^-1*c^-1*b*c >

Outputs are reduced to minimal generators and relations:

Sage

sage: groups.presentation.FGAbelian([3,5,2,7,3])
Finitely presented group < a, b | a^3, b^210, a^-1*b^-1*a*b >
sage: groups.presentation.FGAbelian([3,210])
Finitely presented group < a, b | a^3, b^210, a^-1*b^-1*a*b >

Python

>>> from sage.all import *
>>> groups.presentation.FGAbelian([Integer(3),Integer(5),Integer(2),Integer(7),Integer(3)])
Finitely presented group < a, b | a^3, b^210, a^-1*b^-1*a*b >
>>> groups.presentation.FGAbelian([Integer(3),Integer(210)])
Finitely presented group < a, b | a^3, b^210, a^-1*b^-1*a*b >

The trivial group is an acceptable output:

Sage

sage: groups.presentation.FGAbelian([])
Finitely presented group <  |  >
sage: groups.presentation.FGAbelian([1])
Finitely presented group <  |  >
sage: groups.presentation.FGAbelian([1,1,1,1,1,1,1,1,1,1])
Finitely presented group <  |  >

Python

>>> from sage.all import *
>>> groups.presentation.FGAbelian([])
Finitely presented group <  |  >
>>> groups.presentation.FGAbelian([Integer(1)])
Finitely presented group <  |  >
>>> groups.presentation.FGAbelian([Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(1)])
Finitely presented group <  |  >

Input list must consist of positive integers:

Sage

sage: groups.presentation.FGAbelian([2,6,3,9,-4])
Traceback (most recent call last):
...
ValueError: input list must contain nonnegative entries
sage: groups.presentation.FGAbelian([2,'a',4])
Traceback (most recent call last):
...
TypeError: unable to convert 'a' to an integer

Python

>>> from sage.all import *
>>> groups.presentation.FGAbelian([Integer(2),Integer(6),Integer(3),Integer(9),-Integer(4)])
Traceback (most recent call last):
...
ValueError: input list must contain nonnegative entries
>>> groups.presentation.FGAbelian([Integer(2),'a',Integer(4)])
Traceback (most recent call last):
...
TypeError: unable to convert 'a' to an integer

sage.groups.finitely_presented_named.FinitelyGeneratedHeisenbergPresentation(n=1, p=0)[source]#

Return a finite presentation of the Heisenberg group.

The Heisenberg group is the group of \((n+2) \times (n+2)\) matrices over a ring \(R\) with diagonal elements equal to 1, first row and last column possibly nonzero, and all the other entries equal to zero.

INPUT:

n – the degree of the Heisenberg group
p – (optional) a prime number, where we construct the Heisenberg group over the finite field \(\ZZ/p\ZZ\)

OUTPUT: finitely generated Heisenberg group over the finite field of order p or over the integers

See also

HeisenbergGroup

EXAMPLES:

Sage

sage: H = groups.presentation.Heisenberg(); H
Finitely presented group < x1, y1, z |
 x1*y1*x1^-1*y1^-1*z^-1, z*x1*z^-1*x1^-1, z*y1*z^-1*y1^-1 >
sage: H.order()
+Infinity
sage: r1, r2, r3 = H.relations()
sage: A = matrix([[1, 1, 0], [0, 1, 0], [0, 0, 1]])
sage: B = matrix([[1, 0, 0], [0, 1, 1], [0, 0, 1]])
sage: C = matrix([[1, 0, 1], [0, 1, 0], [0, 0, 1]])
sage: r1(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
sage: r2(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
sage: r3(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
sage: p = 3
sage: Hp = groups.presentation.Heisenberg(p=3)
sage: Hp.order() == p**3
True
sage: Hnp = groups.presentation.Heisenberg(n=2, p=3)
sage: len(Hnp.relations())
13

Python

>>> from sage.all import *
>>> H = groups.presentation.Heisenberg(); H
Finitely presented group < x1, y1, z |
 x1*y1*x1^-1*y1^-1*z^-1, z*x1*z^-1*x1^-1, z*y1*z^-1*y1^-1 >
>>> H.order()
+Infinity
>>> r1, r2, r3 = H.relations()
>>> A = matrix([[Integer(1), Integer(1), Integer(0)], [Integer(0), Integer(1), Integer(0)], [Integer(0), Integer(0), Integer(1)]])
>>> B = matrix([[Integer(1), Integer(0), Integer(0)], [Integer(0), Integer(1), Integer(1)], [Integer(0), Integer(0), Integer(1)]])
>>> C = matrix([[Integer(1), Integer(0), Integer(1)], [Integer(0), Integer(1), Integer(0)], [Integer(0), Integer(0), Integer(1)]])
>>> r1(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
>>> r2(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
>>> r3(A, B, C)
[1 0 0]
[0 1 0]
[0 0 1]
>>> p = Integer(3)
>>> Hp = groups.presentation.Heisenberg(p=Integer(3))
>>> Hp.order() == p**Integer(3)
True
>>> Hnp = groups.presentation.Heisenberg(n=Integer(2), p=Integer(3))
>>> len(Hnp.relations())
13

REFERENCES:

Wikipedia article Heisenberg_group

sage.groups.finitely_presented_named.KleinFourPresentation()[source]#

Build the Klein group of order \(4\) as a finitely presented group.

OUTPUT: Klein four group (\(C_2 \times C_2\)) as a finitely presented group

EXAMPLES:

Sage

sage: K = groups.presentation.KleinFour(); K
Finitely presented group < a, b | a^2, b^2, a^-1*b^-1*a*b >

Python

>>> from sage.all import *
>>> K = groups.presentation.KleinFour(); K
Finitely presented group < a, b | a^2, b^2, a^-1*b^-1*a*b >

sage.groups.finitely_presented_named.QuaternionPresentation()[source]#

Build the Quaternion group of order 8 as a finitely presented group.

OUTPUT: Quaternion group as a finite presentation

EXAMPLES:

Sage

sage: Q = groups.presentation.Quaternion(); Q
Finitely presented group < a, b | a^4, b^2*a^-2, a*b*a*b^-1 >
sage: Q.as_permutation_group().is_isomorphic(QuaternionGroup())
True

Python

>>> from sage.all import *
>>> Q = groups.presentation.Quaternion(); Q
Finitely presented group < a, b | a^4, b^2*a^-2, a*b*a*b^-1 >
>>> Q.as_permutation_group().is_isomorphic(QuaternionGroup())
True

sage.groups.finitely_presented_named.SymmetricPresentation(n)[source]#

Build the Symmetric group of order \(n!\) as a finitely presented group.

INPUT:

n – the size of the underlying set of arbitrary symbols being acted on by the Symmetric group of order \(n!\)

OUTPUT:

Symmetric group as a finite presentation, implementation uses GAP to find an isomorphism from a permutation representation to a finitely presented group representation. Due to this fact, the exact output presentation may not be the same for every method call on a constant n.

EXAMPLES:

Sage

sage: S4 = groups.presentation.Symmetric(4)
sage: S4.as_permutation_group().is_isomorphic(SymmetricGroup(4))
True

Python

>>> from sage.all import *
>>> S4 = groups.presentation.Symmetric(Integer(4))
>>> S4.as_permutation_group().is_isomorphic(SymmetricGroup(Integer(4)))
True