o à8VaÜã@s^ddlmZddlmZddlmZejZdd„Zdd„Zdd „Z d d „Z d d „Z dd„Z dS)é)Ú DirectProduct)ÚPermutationGroup)Ú PermutationcGsRg}d}d}|D]}||7}||9}| t|ƒ¡qt|Ž}d|_||_||_|S)a¯ 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 ========== .. [1] http://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups réT)ÚappendÚ CyclicGrouprÚ _is_abelianÚ_degreeÚ_order)Z cyclic_ordersÚgroupsZdegreeZorderÚsizeÚG©rúB/usr/lib/python3/dist-packages/sympy/combinatorics/named_groups.pyÚ AbelianGroups!rcCs|dvr ttdgƒgƒStt|ƒƒ}|d|d|d|d<|d<|d<|}|dr;ttd|ƒƒ}| d¡|}nttd|ƒƒ}| d¡| dd¡|}||g}||kr]|dd…}tdd„|Dƒdd }|d krsd |_d |_nd|_d|_|d krd |_nd|_||_ d |_ d |_ |S) a= 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 See Also ======== SymmetricGroup, CyclicGroup, DihedralGroup References ========== .. [1] Armstrong, M. "Groups and Symmetry" )rérrrNcSsg|]}t|ƒ‘qSr)Ú_af_new)Ú.0ÚarrrÚ psz$AlternatingGroup..F)ZdupséTé) rrÚlistÚrangerÚinsertrÚ _is_nilpotentÚ _is_solvabler Ú_is_transitiveZ_is_alt)ÚnrÚgen1Úgen2Zgensr rrrÚAlternatingGroup8s:& (    r!cCsRttd|ƒƒ}| d¡t|ƒ}t|gƒ}d|_d|_d|_||_d|_ ||_ |S)a» Generates the cyclic group of order ``n`` as a permutation group. Explanation =========== The generator taken is the ``n``-cycle ``(0 1 2 ... n-1)`` (in cycle notation). After the group is generated, some of its basic properties are set. Examples ======== >>> from sympy.combinatorics.named_groups import CyclicGroup >>> G = CyclicGroup(6) >>> G.is_group True >>> G.order() 6 >>> list(G.generate_schreier_sims(af=True)) [[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] See Also ======== SymmetricGroup, DihedralGroup, AlternatingGroup rrT) rrrrrrrrr rr )rrÚgenr rrrr‚s  rcCsÔ|dkr ttddgƒgƒS|dkr$ttgd¢ƒtgd¢ƒtgd¢ƒgƒSttd|ƒƒ}| d¡t|ƒ}tt|ƒƒ}| ¡t|ƒ}t||gƒ}||d@dkrTd|_nd|_d|_d|_ ||_ d|_ d||_ |S) a€ 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]]] See Also ======== SymmetricGroup, CyclicGroup, AlternatingGroup References ========== .. [1] https://en.wikipedia.org/wiki/Dihedral_group rrr)rrér)rr#rr)r#rrrTF) rrrrrrÚreverserrrr rr )rrrr r rrrÚ DihedralGroup®s,) ÿ    r%cCsÞ|dkr ttdgƒgƒ}n;|dkrttddgƒgƒ}n-ttd|ƒƒ}| d¡t|ƒ}tt|ƒƒ}|d|d|d<|d<t|ƒ}t||gƒ}|dkrSd|_d|_nd|_d|_|dkrad|_nd|_||_ d|_ d|_ |S)aL 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 ======== CyclicGroup, DihedralGroup, AlternatingGroup References ========== .. [1] https://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations rrrr#TFr) rrrrrrrrrr rZ_is_sym)rr rrr rrrÚSymmetricGroupñs.'   r&cCs(ddlm}|dkrtdƒ‚t||ƒƒS)z—Return a group of Rubik's cube generators >>> from sympy.combinatorics.named_groups import RubikGroup >>> RubikGroup(2).is_group True r)Úrubikrz(Invalid cube. n has to be greater than 1)Zsympy.combinatorics.generatorsr'Ú ValueErrorr)rr'rrrÚ RubikGroup4s  r)N) Z$sympy.combinatorics.group_constructsrZsympy.combinatorics.perm_groupsrZ sympy.combinatorics.permutationsrrrr!rr%r&r)rrrrÚs  0J,C C