+ i r^RIHt^RIHt^RIHt]P tRtRtRt Rt Rt R t R t R tR #) ) DirectProduct)PermutationGroup) Permutationc.p^p^pVF-pW$, pW4,pVP\V44K/ \V!pRVnW%nW5nV#)aX Returns the direct product of cyclic groups with the given orders. Explanation =========== According to the structure theorem for finite abelian groups ([1]), every finite abelian group can be written as the direct product of finitely many cyclic groups. Examples ======== >>> from sympy.combinatorics.named_groups import AbelianGroup >>> AbelianGroup(3, 4) PermutationGroup([ (6)(0 1 2), (3 4 5 6)]) >>> _.is_group True See Also ======== DirectProduct References ========== .. [1] https://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups T)append CyclicGroupr _is_abelian_degree_order) cyclic_ordersgroupsdegreeordersizeGs* ^/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/combinatorics/named_groups.py AbelianGrouprs_BF F E  k$'( vAAMIH Hc:VR9d\\^.4.4#\\V44pV^,V^,V^,uV^&V^&V^&TpV^,'d*\\^V44pVP ^4TpM:\\^V44pVP ^4VP ^^4TpW#.pW#8Xd VR,p\VUu.uFp\ V4NK upRR7p\WPV4RVnV#uupi)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" :NNF)dupsT)r) rrlistrangerinsert_af_new set_alternating_group_properties_is_alt)nagen1gen2gensrs& rAlternatingGroupr$8sL F{aS!1 233 U1XAtQqT1Q4AaD!A$! D1uu 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 T) rrrrrr r'r(r r)r r*)rr genrs& rrrsk< U1a[AHHQK !*C#AAMAOANIAH1fAN Hrc,V^8Xd\\^^.4.4#V^8Xd0\\.RO4\.RO4\.RO4.4#\\^V44pVP ^4\ V4p\\V44pVP 4\ V4p\W#.4pW^, ,^8Xd RVnMRVnRVnRVn RVn Wn RVn ^V,Vn V#)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 TF)rrr)rr/rr)r/rrr)rrrrrrreverser'r*r r(r r)r )rr r!r"rs& r DihedralGroupr1sR AvaV!4 566Av\!:<(+l*C!EF F U1a[AHHQK 1:D U1XAIIK 1:D$&AaCyA~ANAMANIAsAH HrcV^8Xd\\^.4.4pMV^8Xd\\^^.4.4pMu\\^V44pVP ^4\ V4p\\V44pV^,V^,uV^&V^&\ V4p\W4.4p\ WV4RVnV#)a Generates the symmetric group on ``n`` elements as a permutation group. Explanation =========== The generators taken are the ``n``-cycle ``(0 1 2 ... n-1)`` and the transposition ``(0 1)`` (in cycle notation). (See [1]). After the group is generated, some of its basic properties are set. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> G = SymmetricGroup(4) >>> G.is_group True >>> G.order() 24 >>> list(G.generate_schreier_sims(af=True)) [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 1, 2, 0], [0, 2, 3, 1], [1, 3, 0, 2], [2, 0, 1, 3], [3, 2, 0, 1], [0, 3, 1, 2], [1, 0, 2, 3], [2, 1, 3, 0], [3, 0, 1, 2], [0, 1, 3, 2], [1, 2, 0, 3], [2, 3, 1, 0], [3, 1, 0, 2], [0, 2, 1, 3], [1, 3, 2, 0], [2, 0, 3, 1], [3, 2, 1, 0], [0, 3, 2, 1], [1, 0, 3, 2], [2, 1, 0, 3], [3, 0, 2, 1]] See Also ======== CyclicGroup, DihedralGroup, AlternatingGroup References ========== .. [1] https://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations T)rrrrrrset_symmetric_group_properties_is_sym)rrr r!r"s& rSymmetricGroupr5sN Av k1#./ 0 a k1a&12 3 q!   qz qNqT1Q4 !adqz d\ *"1+AI HrcV^8dRVnRVnMRVnRVnV^8d RVnMRVnW nRVnVR9VnR#)z+Set known properties of a symmetric group. TFN)rr/r&r+s&&&rr3r31sR1u  1uIA6kANrcT^RIHpV^8:d \R4h\V!V44#)zReturn a group of Rubik's cube generators >>> from sympy.combinatorics.named_groups import RubikGroup >>> RubikGroup(2).is_group True )rubikz(Invalid cube. n has to be greater than 1)sympy.combinatorics.generatorsr8 ValueErrorr)rr8s& r RubikGroupr;Bs)5AvCDD E!H %%rN)$sympy.combinatorics.group_constructsrsympy.combinatorics.perm_groupsr sympy.combinatorics.permutationsrrrr$rrr1r5r3r;rrr@sG><8   - `< ~"* ZA H5 p#" &r