U —9%eº ã@snddlmZddlmZddlmZejZdd„Zdd„Zdd „Z d d „Z d d „Z dd„Z dd„Z dd„ZdS)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] https://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups réT)ÚappendÚ CyclicGrouprÚ _is_abelianÚ_degreeÚ_order)Z cyclic_ordersÚgroupsÚdegreeÚorderÚsizeÚG©rú_/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/combinatorics/named_groups.pyÚ AbelianGroups!rcCsæ|dkrttdgƒgƒStt|ƒƒ}|d|d|d|d<|d<|d<|}|drvttd|ƒƒ}| d¡|}n(ttd|ƒƒ}| d¡| dd¡|}||g}||krº|dd…}tdd„|Dƒdd }t|||ƒd |_|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$AlternatingGroup..F)ZdupsT)rrÚlistÚrangerÚinsertÚ set_alternating_group_propertiesZ_is_alt)ÚnrÚgen1Úgen2ZgensrrrrÚAlternatingGroup8s(& (     rcCsN|dkrd|_d|_n d|_d|_|dkr2d|_nd|_||_d|_d|_dS)z.Set known properties of an alternating group. éTFéN©rÚ _is_nilpotentÚ _is_solvabler Ú_is_transitiveÚ _is_dihedral©rrr rrrrwsrcCs\ttd|ƒƒ}| d¡t|ƒ}t|gƒ}d|_d|_d|_||_d|_ ||_ |dk|_ |S)a» Generates the cyclic group of order ``n`` as a permutation group. Explanation =========== The generator taken is the ``n``-cycle ``(0 1 2 ... n-1)`` (in cycle notation). After the group is generated, some of its basic properties are set. Examples ======== >>> from sympy.combinatorics.named_groups import CyclicGroup >>> G = CyclicGroup(6) >>> G.is_group True >>> G.order() 6 >>> list(G.generate_schreier_sims(af=True)) [[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] See Also ======== SymmetricGroup, DihedralGroup, AlternatingGroup rrTr) rrrrrrr#r$r r%r r&)rrÚgenrrrrrˆs   rcCsæ|dkrttddgƒgƒS|dkrTttddddgƒtddddgƒtddddgƒgƒSttd|ƒƒ}| d¡t|ƒ}tt|ƒƒ}| ¡t|ƒ}t||gƒ}||d@dkr´d|_nd|_d|_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 rrréTF)rrrrrrÚreverser#r&rr$r r%r )rrrrrrrrÚ DihedralGroupµs0)ÿ    r+cCs¦|dkrttdgƒgƒ}nv|dkr6ttddgƒgƒ}nZttd|ƒƒ}| d¡t|ƒ}tt|ƒƒ}|d|d|d<|d<t|ƒ}t||gƒ}t|||ƒd|_|S)aL Generates the symmetric group on ``n`` elements as a permutation group. Explanation =========== The generators taken are the ``n``-cycle ``(0 1 2 ... n-1)`` and the transposition ``(0 1)`` (in cycle notation). (See [1]). After the group is generated, some of its basic properties are set. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> G = SymmetricGroup(4) >>> G.is_group True >>> G.order() 24 >>> list(G.generate_schreier_sims(af=True)) [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 1, 2, 0], [0, 2, 3, 1], [1, 3, 0, 2], [2, 0, 1, 3], [3, 2, 0, 1], [0, 3, 1, 2], [1, 0, 2, 3], [2, 1, 3, 0], [3, 0, 1, 2], [0, 1, 3, 2], [1, 2, 0, 3], [2, 3, 1, 0], [3, 1, 0, 2], [0, 2, 1, 3], [1, 3, 2, 0], [2, 0, 3, 1], [3, 2, 1, 0], [0, 3, 2, 1], [1, 0, 3, 2], [2, 1, 0, 3], [3, 0, 2, 1]] See Also ======== CyclicGroup, DihedralGroup, AlternatingGroup References ========== .. [1] https://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations rrrT)rrrrrrÚset_symmetric_group_propertiesZ_is_sym)rrrrrrrrÚSymmetricGroupùs'    r-cCsR|dkrd|_d|_n d|_d|_|dkr2d|_nd|_||_d|_|dk|_dS)z+Set known properties of a symmetric group. r)TFr!)rr)Nr"r'rrrr,1sr,cCs(ddlm}|dkrtdƒ‚t||ƒƒS)z—Return a group of Rubik's cube generators >>> from sympy.combinatorics.named_groups import RubikGroup >>> RubikGroup(2).is_group True r)Úrubikrz(Invalid cube. n has to be greater than 1)Zsympy.combinatorics.generatorsr.Ú ValueErrorr)rr.rrrÚ RubikGroupBs r0N)Z$sympy.combinatorics.group_constructsrZsympy.combinatorics.perm_groupsrZ sympy.combinatorics.permutationsrrrrrrr+r-r,r0rrrrÚs   0?-D8