Named Groups¶

sympy.combinatorics.named_groups.
SymmetricGroup
(n)[source]¶ Generates the symmetric group on
n
elements as a permutation group.The generators taken are the
n
cycle(0 1 2 ... n1)
and the transposition(0 1)
(in cycle notation). (See [1]). After the group is generated, some of its basic properties are set.See also
References
[1] http://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations
Examples
>>> from sympy.combinatorics.named_groups import SymmetricGroup >>> G = SymmetricGroup(4) >>> G.is_group True >>> G.order() 24 >>> list(G.generate_schreier_sims(af=True)) [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 1, 2, 0], [0, 2, 3, 1], [1, 3, 0, 2], [2, 0, 1, 3], [3, 2, 0, 1], [0, 3, 1, 2], [1, 0, 2, 3], [2, 1, 3, 0], [3, 0, 1, 2], [0, 1, 3, 2], [1, 2, 0, 3], [2, 3, 1, 0], [3, 1, 0, 2], [0, 2, 1, 3], [1, 3, 2, 0], [2, 0, 3, 1], [3, 2, 1, 0], [0, 3, 2, 1], [1, 0, 3, 2], [2, 1, 0, 3], [3, 0, 2, 1]]

sympy.combinatorics.named_groups.
CyclicGroup
(n)[source]¶ Generates the cyclic group of order
n
as a permutation group.The generator taken is the
n
cycle(0 1 2 ... n1)
(in cycle notation). After the group is generated, some of its basic properties are set.See also
Examples
>>> from sympy.combinatorics.named_groups import CyclicGroup >>> G = CyclicGroup(6) >>> G.is_group True >>> G.order() 6 >>> list(G.generate_schreier_sims(af=True)) [[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]

sympy.combinatorics.named_groups.
DihedralGroup
(n)[source]¶ Generates the dihedral group \(D_n\) as a permutation group.
The dihedral group \(D_n\) is the group of symmetries of the regular
n
gon. The generators taken are then
cyclea = (0 1 2 ... n1)
(a rotation of then
gon) andb = (0 n1)(1 n2)...
(a reflection of then
gon) in cycle rotation. It is easy to see that these satisfya**n = b**2 = 1
andbab = ~a
so they indeed generate \(D_n\) (See [1]). After the group is generated, some of its basic properties are set.See also
References
[1] http://en.wikipedia.org/wiki/Dihedral_group
Examples
>>> from sympy.combinatorics.named_groups import DihedralGroup >>> G = DihedralGroup(5) >>> G.is_group True >>> a = list(G.generate_dimino()) >>> [perm.cyclic_form for perm in a] [[], [[0, 1, 2, 3, 4]], [[0, 2, 4, 1, 3]], [[0, 3, 1, 4, 2]], [[0, 4, 3, 2, 1]], [[0, 4], [1, 3]], [[1, 4], [2, 3]], [[0, 1], [2, 4]], [[0, 2], [3, 4]], [[0, 3], [1, 2]]]

sympy.combinatorics.named_groups.
AlternatingGroup
(n)[source]¶ Generates the alternating group on
n
elements as a permutation group.For
n > 2
, the generators taken are(0 1 2), (0 1 2 ... n1)
forn
odd and(0 1 2), (1 2 ... n1)
forn
even (See [1], p.31, ex.6.9.). After the group is generated, some of its basic properties are set. The casesn = 1, 2
are handled separately.See also
References
[1] Armstrong, M. “Groups and Symmetry”
Examples
>>> from sympy.combinatorics.named_groups import AlternatingGroup >>> G = AlternatingGroup(4) >>> G.is_group True >>> a = list(G.generate_dimino()) >>> len(a) 12 >>> all(perm.is_even for perm in a) True

sympy.combinatorics.named_groups.
AbelianGroup
(*cyclic_orders)[source]¶ Returns the direct product of cyclic groups with the given orders.
According to the structure theorem for finite abelian groups ([1]), every finite abelian group can be written as the direct product of finitely many cyclic groups.
See also
DirectProduct
References
[1] http://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups
Examples
>>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.named_groups import AbelianGroup >>> AbelianGroup(3, 4) PermutationGroup([ (6)(0 1 2), (3 4 5 6)]) >>> _.is_group True