ó ¡¼™\c@ s™ddlmZmZddlmZddlmZddlmZddl m Z ej Z d„Z d„Z d„Zd „Zd „Zd „Zd S( iÿÿÿÿ(tprint_functiontdivision(t DirectProduct(tPermutationGroup(t Permutation(trangecG sug}d}d}x5|D]-}||7}||9}|jt|ƒƒqWt|Œ}t|_||_||_|S(sê 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. 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 See Also ======== DirectProduct References ========== .. [1] http://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups ii(tappendt CyclicGroupRtTruet _is_abeliant_degreet_order(t cyclic_orderstgroupstdegreetordertsizetG((s?lib/python2.7/site-packages/sympy/combinatorics/named_groups.pyt AbelianGroup s        cC s–|dkr"ttdgƒgƒStt|ƒƒ}|d|d|d|d<|d<|d<|}|dr›ttd|ƒƒ}|jdƒ|}n8ttd|ƒƒ}|jdƒ|jddƒ|}||g}||krø|d }ntg|D]}t|ƒ^qdtƒ}|dkrDt|_ t|_ nt|_ t|_ |dkrnt|_ n t|_ ||_ t|_ t|_|S(s Generates the alternating group on ``n`` elements as a permutation group. 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" iiitdupsii(ii(RRtlistRRtinsertt_af_newtFalseRR t _is_nilpotentt _is_solvableR t_is_transitivet_is_alt(tntatgen1tgen2tgensR((s?lib/python2.7/site-packages/sympy/combinatorics/named_groups.pytAlternatingGroup:s:# ,       +           cC swttd|ƒƒ}|jdƒt|ƒ}t|gƒ}t|_t|_t|_||_ t|_ ||_ |S(sš Generates the cyclic group of order ``n`` as a permutation group. 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 ii( RRRRRRR RRR RR (RRtgenR((s?lib/python2.7/site-packages/sympy/combinatorics/named_groups.pyRs        cC s@|dkr%ttddgƒgƒS|dkrzttddddgƒtddddgƒtddddgƒgƒSttd|ƒƒ}|jdƒt|ƒ}tt|ƒƒ}|jƒt|ƒ}t||gƒ}||d@dkrt|_n t |_t |_ t|_ ||_ t|_ d||_|S(s\ 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 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 iiii(RRRRRRtreverseRRRR RR RR (RRRRR((s?lib/python2.7/site-packages/sympy/combinatorics/named_groups.pyt DihedralGroupªs*&  1           cC s?|dkr'ttdgƒgƒ}n¥|dkrQttddgƒgƒ}n{ttd|ƒƒ}|jdƒt|ƒ}tt|ƒƒ}|d|d|d<|d>> 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 iiiii(RRRRRRRR RRRR Rt_is_sym(RRRRR((s?lib/python2.7/site-packages/sympy/combinatorics/named_groups.pytSymmetricGroupês.$                cC s;ddlm}|dkr+tdƒ‚nt||ƒƒS(s—Return a group of Rubik's cube generators >>> from sympy.combinatorics.named_groups import RubikGroup >>> RubikGroup(2).is_group True iÿÿÿÿ(trubikis(Invalid cube. n has to be greater than 1(tsympy.combinatorics.generatorsR't ValueErrorR(RR'((s?lib/python2.7/site-packages/sympy/combinatorics/named_groups.pyt RubikGroup*s N(t __future__RRt$sympy.combinatorics.group_constructsRtsympy.combinatorics.perm_groupsRt sympy.combinatorics.permutationsRtsympy.core.compatibilityRRRR!RR$R&R*(((s?lib/python2.7/site-packages/sympy/combinatorics/named_groups.pyts  / G ) @ @