g hddlmZddlmZddlmZejZdZdZdZ dZ dZ d Z d Z d Zd S) ) DirectProduct)PermutationGroup) Permutationcg}d}d}|D].}||z }||z}|t|/t|}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 rT)append CyclicGroupr _is_abelian_degree_order) cyclic_ordersgroupsdegreeordersizeGs \/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/combinatorics/named_groups.py AbelianGrouprsvBF F E))$   k$''((((vAAMAIAH Hc`|dvrttdggStt|}|d|d|dc|d<|d<|d<|}|dzr5ttd|}|d|}nJttd|}|d|dd|}||g}||kr |dd}td|Dd}t |||d |_|S) a= Generates the alternating group on ``n`` elements as a permutation group. Explanation =========== For ``n > 2``, the generators taken are ``(0 1 2), (0 1 2 ... n-1)`` for ``n`` odd and ``(0 1 2), (1 2 ... n-1)`` for ``n`` even (See [1], p.31, ex.6.9.). After the group is generated, some of its basic properties are set. The cases ``n = 1, 2`` are handled separately. Examples ======== >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> G = AlternatingGroup(4) >>> G.is_group True >>> a = list(G.generate_dimino()) >>> len(a) 12 >>> all(perm.is_even for perm in a) True See Also ======== SymmetricGroup, CyclicGroup, DihedralGroup References ========== .. [1] Armstrong, M. "Groups and Symmetry" )rrrrNc,g|]}t|S)_af_new).0as r z$AlternatingGroup..ps333'!**333rF)dupsT)rrlistrangerinsert set_alternating_group_properties_is_alt)nrgen1gen2gensrs rAlternatingGroupr(8s6L F{{aS!1!1 2333 U1XXAtQqT1Q4AaD!A$! D1u q!     q!     A $>> from sympy.combinatorics.named_groups import CyclicGroup >>> G = CyclicGroup(6) >>> G.is_group True >>> G.order() 6 >>> list(G.generate_schreier_sims(af=True)) [[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] See Also ======== SymmetricGroup, DihedralGroup, AlternatingGroup rrTr) rr rrrr r-r.r r/r r0)r$rgenrs rr r s}< U1a[[AHHQKKK !**C#AAMAOANAIAAH1fAN Hrc|dkrttddggS|dkr?ttgdtgdtgdgSttd|}|dt |}tt|}|t |}t||g}||dz zdkrd|_nd|_d|_d|_ d|_ ||_ d|_ d|z|_ |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)rrr)rr5rr)r5rrrTF)rrrr rrreverser-r0r r.r r/r )r$rr%r&rs r DihedralGroupr7s:R AvvaV!4!4 5666Avv\\\!:!:<<<((+lll*C*C!EFF F U1a[[AHHQKKK 1::D U1XXAIIKKK 1::D$&&AAaCyA~~ANAMANAIAsAH Hrc|dkrttdgg}n|dkr ttddgg}nttd|}|dt |}tt|}|d|dc|d<|d<t |}t||g}t |||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 rrrT)rrrr rrset_symmetric_group_properties_is_sym)r$rrr%r&s rSymmetricGroupr;sN Avv k1#../ 0 0 a k1a&112 3 3 q!    qzz qNNqT1Q4 !adqzz dD\ * *"1a+++AI Hrc|dkrd|_d|_nd|_d|_|dkrd|_nd|_||_d|_|dv|_dS)z+Set known properties of a symmetric group. r5TFr+)rr5Nr,r1s rr9r91s`1uu  1uuAIA6kANNNrchddlm}|dkrtdt||S)zReturn a group of Rubik's cube generators >>> from sympy.combinatorics.named_groups import RubikGroup >>> RubikGroup(2).is_group True r)rubikrz(Invalid cube. n has to be greater than 1)sympy.combinatorics.generatorsr> ValueErrorr)r$r>s r RubikGrouprABsD544444AvvCDDD EE!HH % %%rN)$sympy.combinatorics.group_constructsrsympy.combinatorics.perm_groupsr sympy.combinatorics.permutationsrrrr(r"r r7r;r9rArrrrEs>>>>>><<<<<<888888  - - - `< < < ~"* * * ZA A A H5 5 5 p###" & & & & &r