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]]
See also
References
- 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]]
See also
- 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 then
-cyclea = (0 1 2 ... n-1)
(a rotation of then
-gon) andb = (0 n-1)(1 n-2)...
(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.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]]]
See also
References
- 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)
forn
odd and(0 1 2), (1 2 ... n-1)
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.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
See also
References
- R44
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
References