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 [THOMAS-WOODS].
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
- 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 >
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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)
TESTS:
sage: #even permutation test.. sage: A1 = groups.presentation.Alternating(1); A2 = groups.presentation.Alternating(2) sage: A1.is_isomorphic(A2), A1.order() (True, 1) sage: A3 = groups.presentation.Alternating(3); A3.order(), A3.as_permutation_group().is_cyclic() (3, True) sage: A8 = groups.presentation.Alternating(8); A8.order() 20160
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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 [Sun]):
\[BD_n = \langle x, y, z | x^2 = y^2 = z^n = x y z \rangle.\]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 >
TESTS:
sage: for n in range(3, 9): ....: P = groups.presentation.BinaryDihedral(n) ....: M = groups.matrix.BinaryDihedral(n) ....: assert P.is_isomorphic(M)
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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
TESTS:
sage: groups.presentation.Cyclic(0) Traceback (most recent call last): ... ValueError: finitely presented group order must be positive
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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
TESTS:
sage: Q = groups.presentation.DiCyclic(2) sage: Q.as_permutation_group().is_isomorphic(QuaternionGroup()) True sage: all([groups.presentation.DiCyclic(i).as_permutation_group( ....: ).is_isomorphic(groups.permutation.DiCyclic(i)) for i in [5,8,12,2^5]]) True sage: groups.presentation.DiCyclic(1) Traceback (most recent call last): ... ValueError: input integer must be greater than 1
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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
TESTS:
sage: n = 9 sage: D = groups.presentation.Dihedral(n) sage: D.ngens() == 2 True sage: groups.presentation.Dihedral(0) Traceback (most recent call last): ... ValueError: finitely presented group order must be positive
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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
TESTS:
sage: ag = groups.presentation.FGAbelian([2,2]) sage: ag.as_permutation_group().is_isomorphic(groups.permutation.KleinFour()) True sage: G = groups.presentation.FGAbelian([2,4,8]) sage: C2 = CyclicPermutationGroup(2) sage: C4 = CyclicPermutationGroup(4) sage: C8 = CyclicPermutationGroup(8) sage: gg = (C2.direct_product(C4)[0]).direct_product(C8)[0] sage: gg.is_isomorphic(G.as_permutation_group()) True sage: all([groups.presentation.FGAbelian([i]).as_permutation_group().is_isomorphic(groups.presentation.Cyclic(i).as_permutation_group()) for i in [2..35]]) True
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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 >
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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
TESTS:
sage: Q = groups.presentation.Quaternion() sage: Q.order(), Q.is_abelian() (8, False) sage: Q.is_isomorphic(groups.presentation.DiCyclic(2)) True
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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
TESTS:
sage: S = [groups.presentation.Symmetric(i) for i in range(1,4)]; S[0].order() 1 sage: S[1].order(), S[2].as_permutation_group().is_isomorphic(DihedralGroup(3)) (2, True) sage: S5 = groups.presentation.Symmetric(5) sage: perm_S5 = S5.as_permutation_group(); perm_S5.is_isomorphic(SymmetricGroup(5)) True sage: groups.presentation.Symmetric(8).order() 40320