Named Groups

sympy.combinatorics.named_groups.SymmetricGroup(n)[source]

Generates the symmetric group on n elements as a permutation group.

Explanation

The generators taken are the n-cycle (0 1 2 ... n-1) and the transposition (0 1) (in cycle notation). (See [1]). After the group is generated, some of its basic properties are set.

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]]

References

R43: https://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations

sympy.combinatorics.named_groups.CyclicGroup(n)[source]

Generates the cyclic group of order n as a permutation group.

Explanation

The generator taken is the n-cycle (0 1 2 ... n-1) (in cycle notation). After the group is generated, some of its basic properties are set.

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.

Explanation

The dihedral group \(D_n\) is the group of symmetries of the regular n-gon. The generators taken are the n-cycle a = (0 1 2 ... n-1) (a rotation of the n-gon) and b = (0 n-1)(1 n-2)... (a reflection of the n-gon) in cycle rotation. It is easy to see that these satisfy a**n = b**2 = 1 and bab = ~a so they indeed generate \(D_n\) (See [1]). After the group is generated, some of its basic properties are set.

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]]]

References

R44: https://en.wikipedia.org/wiki/Dihedral_group

sympy.combinatorics.named_groups.AlternatingGroup(n)[source]

Generates the alternating group on n elements as a permutation group.

Explanation

For n > 2, the generators taken are (0 1 2), (0 1 2 ... n-1) for n odd and (0 1 2), (1 2 ... n-1) for n even (See [1], p.31, ex.6.9.). After the group is generated, some of its basic properties are set. The cases n = 1, 2 are handled separately.

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

References

R45: Armstrong, M. “Groups and Symmetry”

sympy.combinatorics.named_groups.AbelianGroup(*cyclic_orders)[source]

Returns the direct product of cyclic groups with the given orders.

Explanation

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.

Examples

>>> from sympy.combinatorics.named_groups import AbelianGroup
>>> AbelianGroup(3, 4)
PermutationGroup([
        (6)(0 1 2),
        (3 4 5 6)])
>>> _.is_group
True

See also

DirectProduct

References

R46: http://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups