U ZʹjgS@sTddlmZddlmZddlmZddlmZGdddeZGdddeZ d S) isprime)PermutationGroup)DefaultPrinting) free_groupc@s.eZdZdZdZd ddZddZddZdS) PolycyclicGroupTNcCs.||_||_||_|s$t|j||n||_dS)a Parameters ========== pc_sequence : list A sequence of elements whose classes generate the cyclic factor groups of pc_series. pc_series : list A subnormal sequence of subgroups where each factor group is cyclic. relative_order : list The orders of factor groups of pc_series. collector : Collector By default, it is None. Collector class provides the polycyclic presentation with various other functionalities. N)pcgs pc_seriesrelative_order Collector collector)selfZ pc_sequencer r r r\/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/combinatorics/pc_groups.py__init__ szPolycyclicGroup.__init__cCstdd|jDS)Ncss|]}t|VqdSNr).0orderrrr $sz1PolycyclicGroup.is_prime_order..)allr r rrris_prime_order#szPolycyclicGroup.is_prime_ordercCs t|jSr)lenrrrrrlength&szPolycyclicGroup.length)N)__name__ __module__ __qualname__Zis_groupZ is_solvablerrrrrrrrs  rc@szeZdZdZdddZddZddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ ddZddZddZdS)r z References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" Section 8.1.3 NcCsX||_||_||_|s,tdt|dn||_ddt|jjD|_| |_ dS)a Most of the parameters for the Collector class are the same as for PolycyclicGroup. Others are described below. Parameters ========== free_group_ : tuple free_group_ provides the mapping of polycyclic generating sequence with the free group elements. pc_presentation : dict Provides the presentation of polycyclic groups with the help of power and conjugate relators. See Also ======== PolycyclicGroup zx:{}rcSsi|]\}}||qSrr)risrrr Osz&Collector.__init__..N) rr r rformatr enumeratesymbolsindex pc_relatorspc_presentation)r rr r Z free_group_r%rrrr5s  zCollector.__init__c Cs|sdS|j}|j}|j}tt|D]F}||\}}|||r&|dks^||||dkr&||ffSq&tt|dD]T}||\}}||d\}} ||||kr~| dkrdnd} ||f|| ffSq~dS)a Returns the minimal uncollected subwords. Explanation =========== A word ``v`` defined on generators in ``X`` is a minimal uncollected subword of the word ``w`` if ``v`` is a subword of ``w`` and it has one of the following form * `v = {x_{i+1}}^{a_j}x_i` * `v = {x_{i+1}}^{a_j}{x_i}^{-1}` * `v = {x_i}^{a_j}` for `a_j` not in `\{1, \ldots, s-1\}`. Where, ``s`` is the power exponent of the corresponding generator. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics import free_group >>> G = SymmetricGroup(4) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> F, x1, x2 = free_group("x1, x2") >>> word = x2**2*x1**7 >>> collector.minimal_uncollected_subword(word) ((x2, 2),) Nr) array_formr r#ranger) r wordarrayrer#rs1e1s2e2errrminimal_uncollected_subwordRs # ( z%Collector.minimal_uncollected_subwordcCsDi}i}|jD](\}}t|jdkr2|||<q|||<q||fS)a Separates the given relators of pc presentation in power and conjugate relations. Returns ======= (power_rel, conj_rel) Separates pc presentation into power and conjugate relations. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> G = SymmetricGroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> power_rel, conj_rel = collector.relations() >>> power_rel {x0**2: (), x1**3: ()} >>> conj_rel {x0**-1*x1*x0: x1**2} See Also ======== pc_relators r&)r%itemsrr()r Zpower_relatorsZconjugate_relatorskeyvaluerrr relationss  zCollector.relationscCsvd}d}tt|t|dD]0}|||t||kr |}|t|}qRq ||krfdkrnnndS||fS)a Returns the start and ending index of a given subword in a word. Parameters ========== word : FreeGroupElement word defined on free group elements for a polycyclic group. w : FreeGroupElement subword of a given word, whose starting and ending index to be computed. Returns ======= (i, j) A tuple containing starting and ending index of ``w`` in the given word. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics import free_group >>> G = SymmetricGroup(4) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> F, x1, x2 = free_group("x1, x2") >>> word = x2**2*x1**7 >>> w = x2**2*x1 >>> collector.subword_index(word, w) (0, 3) >>> w = x1**7 >>> collector.subword_index(word, w) (2, 9) r'r&)r'r')r)rZsubword)r r*wlowhighrrrr subword_indexs( zCollector.subword_indexcCsJ|j}|dd}|dd}|df|df|dff}|j|}|j|S)a Return a conjugate relation. Explanation =========== Given a word formed by two free group elements, the corresponding conjugate relation with those free group elements is formed and mapped with the collected word in the polycyclic presentation. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics import free_group >>> G = SymmetricGroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> F, x0, x1 = free_group("x0, x1") >>> w = x1*x0 >>> collector.map_relation(w) x1**2 See Also ======== pc_presentation rr&r')r(rdtyper%)r r7r+r-r/r4rrr map_relations    zCollector.map_relationcCs |j}||}|sq||||\}}|dkr8q|d\}}t|dkr|j|j|}||} || |} |dd|ff} || } |j| r|j| j} | d\} }|dd| f| | |ff}||}n,| dkr|dd| ff}||}nd}| |||}t|dkr|dddkr|d\}}|dff}||}| ||}|||}||}| |||}qt|dkr|dddkr|d\}}|dff}||}| ||}|d||}||}| |||}q|S)a Return the collected form of a word. Explanation =========== A word ``w`` is called collected, if `w = {x_{i_1}}^{a_1} * \ldots * {x_{i_r}}^{a_r}` with `i_1 < i_2< \ldots < i_r` and `a_j` is in `\{1, \ldots, {s_j}-1\}`. Otherwise w is uncollected. Parameters ========== word : FreeGroupElement An uncollected word. Returns ======= word A collected word of form `w = {x_{i_1}}^{a_1}, \ldots, {x_{i_r}}^{a_r}` with `i_1, i_2, \ldots, i_r` and `a_j \in \{1, \ldots, {s_j}-1\}`. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics import free_group >>> G = SymmetricGroup(4) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> F, x0, x1, x2, x3 = free_group("x0, x1, x2, x3") >>> word = x3*x2*x1*x0 >>> collected_word = collector.collected_word(word) >>> free_to_perm = {} >>> free_group = collector.free_group >>> for sym, gen in zip(free_group.symbols, collector.pcgs): ... free_to_perm[sym] = gen >>> G1 = PermutationGroup() >>> for w in word: ... sym = w[0] ... perm = free_to_perm[sym] ... G1 = PermutationGroup([perm] + G1.generators) >>> G2 = PermutationGroup() >>> for w in collected_word: ... sym = w[0] ... perm = free_to_perm[sym] ... G2 = PermutationGroup([perm] + G2.generators) The two are not identical, but they are equivalent: >>> G1.equals(G2), G1 == G2 (True, False) See Also ======== minimal_uncollected_subword r'rr&N) rr2r:r;rr r#r%r(Zeliminate_wordr<Zsubstituted_word)r r*rr7r8r9r-r.r,qrr4Z presentationsymexpZword_r/r0rrrcollected_word sRA                    zCollector.collected_wordcCs|j}|j}i}i}|j}t||jD] \}}|d||d<|||<q&|ddd}|jddd}|ddd}g} t|D]F\} }|| } ||| } || } | j|| dd}||j }|D]}|||}q| |}|r|nd|| <||_ | |t | dkr| t | d}||}tt | dD]}|| |}|d||} |d| ||}| j|dd}||j }|D]}|||}q| |}|r|nd|| <||_ q:q|S)aM Return the polycyclic presentation. Explanation =========== There are two types of relations used in polycyclic presentation. * Power relations : Power relators are of the form `x_i^{re_i}`, where `i \in \{0, \ldots, \mathrm{len(pcgs)}\}`, ``x`` represents polycyclic generator and ``re`` is the corresponding relative order. * Conjugate relations : Conjugate relators are of the form `x_j^-1x_ix_j`, where `j < i \in \{0, \ldots, \mathrm{len(pcgs)}\}`. Returns ======= A dictionary with power and conjugate relations as key and their collected form as corresponding values. Notes ===== Identity Permutation is mapped with empty ``()``. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.permutations import Permutation >>> S = SymmetricGroup(49).sylow_subgroup(7) >>> der = S.derived_series() >>> G = der[len(der)-2] >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> pcgs = PcGroup.pcgs >>> len(pcgs) 6 >>> free_group = collector.free_group >>> pc_resentation = collector.pc_presentation >>> free_to_perm = {} >>> for s, g in zip(free_group.symbols, pcgs): ... free_to_perm[s] = g >>> for k, v in pc_resentation.items(): ... k_array = k.array_form ... if v != (): ... v_array = v.array_form ... lhs = Permutation() ... for gen in k_array: ... s = gen[0] ... e = gen[1] ... lhs = lhs*free_to_perm[s]**e ... if v == (): ... assert lhs.is_identity ... continue ... rhs = Permutation() ... for gen in v_array: ... s = gen[0] ... e = gen[1] ... rhs = rhs*free_to_perm[s]**e ... assert lhs == rhs r'NToriginalrr&)rr rzip generatorsr r!generator_productreverseidentityrBr%appendrr))r rZ rel_orderr$ perm_to_freergenrZseriesZcollected_gensrr,ZrelationGlr*gZconjZ conjugatorjZ conjugatedgensrrrr$sRC        zCollector.pc_relatorsc Cs|j}t}|jD]}t|g|j}q|j|dd}|i}t|j|jD] \}}|d||d<|||<qP|j}|D]}|||}q|||} |j } dgt |} | j } | D]} | d| | | d<q| S)aJ Return the exponent vector of length equal to the length of polycyclic generating sequence. Explanation =========== For a given generator/element ``g`` of the polycyclic group, it can be represented as `g = {x_1}^{e_1}, \ldots, {x_n}^{e_n}`, where `x_i` represents polycyclic generators and ``n`` is the number of generators in the free_group equal to the length of pcgs. Parameters ========== element : Permutation Generator of a polycyclic group. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.permutations import Permutation >>> G = SymmetricGroup(4) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> pcgs = PcGroup.pcgs >>> collector.exponent_vector(G[0]) [1, 0, 0, 0] >>> exp = collector.exponent_vector(G[1]) >>> g = Permutation() >>> for i in range(len(exp)): ... g = g*pcgs[i]**exp[i] if exp[i] else g >>> assert g == G[1] References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" Section 8.1.1, Definition 8.4 TrCr'rr&) rrrrFrGrHrErIrBr#rr() r elementrrMrOrQrKr@r7r*r# exp_vectortrrrexponent_vectors(-   zCollector.exponent_vectorcCs,||}tddt|Dt|jdS)a Return the depth of a given element. Explanation =========== The depth of a given element ``g`` is defined by `\mathrm{dep}[g] = i` if `e_1 = e_2 = \ldots = e_{i-1} = 0` and `e_i != 0`, where ``e`` represents the exponent-vector. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> G = SymmetricGroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> collector.depth(G[0]) 2 >>> collector.depth(G[1]) 1 References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" Section 8.1.1, Definition 8.5 css|]\}}|r|dVqdS)r&Nr)rrxrrrr`sz"Collector.depth..r&)rUnextr!rr)r rRrSrrrdepth@s zCollector.depthcCs6||}||}|t|jdkr2||dSdS)a  Return the leading non-zero exponent. Explanation =========== The leading exponent for a given element `g` is defined by `\mathrm{leading\_exponent}[g]` `= e_i`, if `\mathrm{depth}[g] = i`. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> G = SymmetricGroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> collector.leading_exponent(G[1]) 1 r&N)rUrXrr)r rRrSrXrrrleading_exponentbs    zCollector.leading_exponentcCs|}||}|t|jkr|||ddkr|||d}||||d}||j|d}|| |}||}q|S)Nr&r')rXrrrYr )r zrOhdkr1rrr_sift}s   zCollector._siftcCsdgt|j}|}|r|d}|||}||}|t|jkr|D]*}|dkrJ||d|d||qJ|||d<qdd|D}|S)a8 Parameters ========== gens : list A list of generators on which polycyclic subgroup is to be defined. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> S = SymmetricGroup(8) >>> G = S.sylow_subgroup(2) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> gens = [G[0], G[1]] >>> ipcgs = collector.induced_pcgs(gens) >>> [gen.order() for gen in ipcgs] [2, 2, 2] >>> G = S.sylow_subgroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> gens = [G[0], G[1]] >>> ipcgs = collector.induced_pcgs(gens) >>> [gen.order() for gen in ipcgs] [3] r&rr'cSsg|]}|dkr|qS)r&r)rrLrrr sz*Collector.induced_pcgs..)rrpopr^rXrJ)r rQrZrMrOr[r\rLrrr induced_pcgss    zCollector.induced_pcgsc Csdgt|}|}||}t|D]^\}}|||kr$||||}||j|d}|| |}|||<||}q,q$|dkr|SdS)z> Return the exponent vector for induced pcgs. rr&F)rrXr!rYr ) r ZipcgsrOr1r[r\rrLfrrrconstructive_membership_tests z&Collector.constructive_membership_test)NN)rrr__doc__rr2r6r:r<rBr$rUrXrYr^rarcrrrrr *s :'3'uyE" -r N) Zsympy.ntheory.primetestrZsympy.combinatorics.perm_groupsrZsympy.printing.defaultsrZsympy.combinatorics.free_groupsrrr rrrrs    #