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: 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.CactusPresentation(n)#
Build the \(n\)-fruit cactus group as a finitely presented group.
OUTPUT:
Cactus group \(J_n\) as a finitely presented group.
EXAMPLES:
sage: J3 = groups.presentation.Cactus(3); J3 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)#
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 groupp
– (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
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:
- 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