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

You can also import the desired functions:

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

sage.groups.finitely_presented_named.AlternatingPresentation(n)

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

sage.groups.finitely_presented_named.BinaryDihedralPresentation(n)

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

sage.groups.finitely_presented_named.CyclicPresentation(n)

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: 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

sage.groups.finitely_presented_named.DiCyclicPresentation(n)

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: 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

sage.groups.finitely_presented_named.DihedralPresentation(n)

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: 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

sage.groups.finitely_presented_named.FinitelyGeneratedAbelianPresentation(int_list)

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: 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 >

You can create free abelian groups:

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 >

And various infinite abelian groups:

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 >

Outputs are reduced to minimal generators and relations:

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 >

The trivial group is an acceptable output:

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 <  |  >

Input list must consist of positive integers:

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

sage.groups.finitely_presented_named.FinitelyGeneratedHeisenbergPresentation(n=1, p=0)

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: 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

REFERENCES:

• Wikipedia article Heisenberg_group

sage.groups.finitely_presented_named.KleinFourPresentation()

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: 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()

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

OUTPUT:

Quaternion group as a finite presentation.

EXAMPLES:

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

sage.groups.finitely_presented_named.SymmetricPresentation(n)

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