o o h @snddlmZddlmZddlmZejZddZddZdd Z d d Z d d Z ddZ ddZ ddZdS)) DirectProduct)PermutationGroup) PermutationcGsRg}d}d}|D]}||7}||9}|t|qt|}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_ordersgroupsdegreeordersizeGrt/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/sympy/combinatorics/named_groups.py AbelianGroups!rcCs|dvr ttdggStt|}|d|d|d|d<|d<|d<|}|dr;ttd|}|d|}nttd|}|d|dd|}||g}||kr]|dd}tdd|Ddd }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" )rrrrNcSsg|]}t|qSr)_af_new).0arrr psz$AlternatingGroup..F)dupsT)rrlistrangerinsert set_alternating_group_properties_is_alt)nrgen1gen2gensrrrrAlternatingGroup8s(& (     r#cCsN|dkr d|_d|_nd|_d|_|dkrd|_nd|_||_d|_d|_dS)z.Set known properties of an alternating group. TFNr _is_nilpotent _is_solvabler _is_transitive _is_dihedralrrr rrrrws rcCs\ttd|}|dt|}t|g}d|_d|_d|_||_d|_ ||_ |dk|_ |S)a Generates the cyclic group of order ``n`` as a permutation group. Explanation =========== The generator taken is the ``n``-cycle ``(0 1 2 ... n-1)`` (in cycle notation). After the group is generated, some of its basic properties are set. Examples ======== >>> from sympy.combinatorics.named_groups import CyclicGroup >>> G = CyclicGroup(6) >>> G.is_group True >>> G.order() 6 >>> list(G.generate_schreier_sims(af=True)) [[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] See Also ======== SymmetricGroup, DihedralGroup, AlternatingGroup rrTr) rrrrrrr'r(r r)r r*)rrgenrrrrrs   rcCs|dkr ttddggS|dkr$ttgdtgdtgdgSttd|}|dt|}tt|}|t|}t||g}||d@dkrTd|_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 rrr)rrr)rr-rr)r-rrrTF)rrrrrrreverser'r*rr(r r)r )rrr r!rrrr DihedralGroups.)     r/cCs|dkr ttdgg}n;|dkrttddgg}n-ttd|}|dt|}tt|}|d|d|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)rrrrrrset_symmetric_group_properties_is_sym)rrrr r!rrrSymmetricGroups'    r2cCsR|dkr d|_d|_nd|_d|_|dkrd|_nd|_||_d|_|dv|_dS)z+Set known properties of a symmetric group. r-TFr%)rr-Nr&r+rrrr01sr0cCs(ddlm}|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.generatorsr3 ValueErrorr)rr3rrr RubikGroupBs  r6N)$sympy.combinatorics.group_constructsrsympy.combinatorics.perm_groupsr sympy.combinatorics.permutationsrrrr#rrr/r2r0r6rrrrs  0?-D8