+ iSb^RIHt^RIHt^RIHt^RIHt!RR]4t!RR]4t R #) isprime)PermutationGroup)DefaultPrinting) free_groupc>a]tRt^toRtRtRRltRtRtRt Vt R#)PolycyclicGroupTNc WnW nW0nV'g\VPW#4VnR#TVnR#)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)self pc_sequencer r rs&&&&&[/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/combinatorics/pc_groups.py__init__PolycyclicGroup.__init__ s/$ ",PY499iH_hc\;QJd&RVP4F 'dK R# R#!RVP44#)c38"TFp\V4xK R#5iNr).0orders& r 1PolycyclicGroup.is_prime_order..$sC/Be75>>/BsFT)allr rs&ris_prime_orderPolycyclicGroup.is_prime_order#s7sCt/B/BCssCsCsCt/B/BCCCrc,\VP4#r)lenr rs&rlengthPolycyclicGroup.length&s499~r)rr r r r) __name__ __module__ __qualname____firstlineno__is_group is_solvablerrr#__static_attributes____classdictcell__ __classdict__s@rr r s%HKi.Drr cva]tRt^*toRtRRltRtRtRtRt Rt R t R t R t R tR tRtRtRtVtR#)rz References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" Section 8.1.3 Nc BWnW nW0nV'g+\RP \ V444^,MTVn\ VPP4UUu/uFwrgWvbK uppVnVP4Vn R#uuppi)a Most of the parameters for the Collector class are the same as for PolycyclicGroup. Others are described below. Parameters ========== free_group_ : tuple free_group_ provides the mapping of polycyclic generating sequence with the free group elements. pc_presentation : dict Provides the presentation of polycyclic groups with the help of power and conjugate relators. See Also ======== PolycyclicGroup zx:{}N) r r r rformatr" enumeratesymbolsindex pc_relatorspc_presentation)rr r r free_group_r6iss&&&&&& rrCollector.__init__5sy, ",IT*V]]3t9%=>qAZe'01H1H'IJ'Itqad'IJ #//1Ks. Bc V'gR#VPpVPpVPp\\ V44FKpW%,wrgW4V,,'gK$V^8gWsWF,,^, 8gKGWg33u# \\ V4^, 4FDpW%,wrgW%^,,wrWF,WH,8gK3V ^8d^MRp Wg3W33u# R#)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),) N) array_formr r4ranger") rwordarrayrer4r8s1e1s2e2es && rminimal_uncollected_subword%Collector.minimal_uncollected_subwordRsF   s5z"AXFB)}}"q&BEIq,@|# # s5z!|$AXFBQ3ZFBy59$aAR2'** %rc /p/pVPP4F)wr4\VP4^8XdWAV&K%WBV&K+ W3#)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 )r6itemsr"r=)rpower_relatorsconjugate_relatorskeyvalues& r relationsCollector.relationssV<..446JC3>>"a'&+s#*/3' 7 11rc RpRp\\V4\V4, ^,4FApVPWU\V4,4V8XgK+TpV\V4,pW43# W43#)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. If not exists, (-1,-1) is returned. 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) >>> w = x1**8 >>> collector.subword_index(word, w) (-1, -1) r<)r>r"subword)rr?wlowhighr8s&&& r subword_indexCollector.subword_indexspVs4yQ')*A||AQx(A-Qxy + yrc VPpV^,^,pV^,^,pVR3V^3V^33pVPPV4pVPV,#)aW 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 r<)r=rdtyper6)rrSr@rBrDrMs&& r map_relationCollector.map_relationsd>  1Xa[ 1Xa[Bx"a2q'*oo##C(##C((rc VPpVPV4pV'gV#VPWP!V44wrEVR8XdKIV^,wrg\ V4^8XEd"VP VP V,,pWx,p WyV,, p V^,^,V33p VP!V 4p VPV ,'dXVPV ,Pp V ^,wrV^,^,V 3WV,33pVP!V4pM/V ^8wd'V^,^,V 33pVP!V4pMRpVPVP!V4V4p\ V4^8XdV^,^,^8d{V^,wppV^33pVP!V4pVPVP!V44pVW,,pVP!V4pVPWEV4pEK%\ V4^8XgEK8V^,^,^8gEKPV^,wppV^33pVP!V4pVPVP!V44pVR,W,,pVP!V4pVPWEV4pEK)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 Nr<) rrGrVrYr"r r4r6r=eliminate_wordrZsubstituted_word)rr?rrSrTrUrBrCrAqrrM presentationsymexpword_rDrEs&& rcollected_wordCollector.collected_wordsB__ 006AZ W**41A1A!1DEICbyqTFB1v{((B8HtG!Q}' &&s+'',,#'#7#7#<#G#GL+AHCd1gq\C3<8E&,,U3EAv"#A$q'1 0 * 0 0 7 $**:+;+;A+>F1v{qtAw{1B1g[%%b)))**:*:1*=>59 "((/,,S>Q11a11B1g[%%b)))**:*:1*=>Buy("((/,,S>rc VPpVPp/p/pVPp\WQP4FwrgVR,WFR,&WtV&K VRRR1,pVP RRR1,pVRRR1,p.p \ V4EFwrW*,p WF,V ,p W,p V PWk,RR7pVP4VPpVFpWV,,pK VPV4pV'dTMRW<&W0n V PV4\V 4^8gKV \V 4^, ,pVV,p\\V 4^, 4FpWIV,,pVR,V,V,p VR,V V,,V,pV PVRR7pVP4VPpVFpWV,,pK VPV4pV'dTMRW<&W0n K EK V#)a 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 NToriginalr<)rr r zip generatorsr r2generator_productreverseidentityrer6appendr"r>)rr rel_orderr5 perm_to_freer genr9seriescollected_gensr8rArelationGlr?gconj conjugatorj conjugatedgenss& rr5Collector.pc_relatorssF__ ''   yy$ 5 56FC$%rELb ! ! 7DbDz"%ddO oFAB#(",H A##CG#=A IIK&&DO+&&t,D,0DbK !#.  ! !# &>"Q&%c.&9!&;<)$/ s>2145A!-Q.?!@J)2~j8CH8N1$55d:D++DT+BAIIK%..D#O3 ..t4D48DbK)+6(6+&Jrc ^VPp\4pVPF p\V.VP,4pK" VP VRR7pVP 4/p\ VPVP4FwrtVR,WdR,&WvV&K VPpVFpWV,,pK VPV4p VPp ^.\V4,p V Pp V Fp V ^,WV ^,,&K V #)a: 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 Trhr<) rrr rlrmrnrkrorer4r"r=) relementrrwryr~rrrbrSr?r4 exp_vectorts && rexponent_vectorCollector.exponent_vectors Z__  A !q||!34A""7t"<  *//;FC"%r'LB !O<   Aq/!A""1% SZ( A&'dJQqT{ #rc VPV4p\R\V44\VP4^,4#)ai 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 c3L"TFwrV'gKV^,xK R#5i)Nrj)rr8xs& rr"Collector.depth..as@%:TQaSQqSS%:s $$)rnextr2r"r )rrrs&& rdepthCollector.depthAs:>))'2 @Yz%:@#dii.QRBRSSrc VPV4pVPV4pV\VP4^,8wdW#^, ,#R#)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 N)rrr"r )rrrrs&& rleading_exponentCollector.leading_exponentcsG*))'2  7# C N1$ $Ag& &rcTpVPV4pV\VP48dW^, ,^8wdW^, ,pVPV4VPV4R,,pW`PV^, ,,pWV),V,pVPV4pKV#)rr<)rr"r rr )rzryhdkrFs&&& r_siftCollector._sift~s  JJqM#dii. QsVq[A#A%%a($*?*?*BR)GGA''!,,A2aA 1 Arc ^.\VP4,pTpV'dVP^4pVPW$4pVP V4pV\VP48gKWVF@pV^8wgK VP VR,VR,,V,V,4KB WRV^, &KVUu.uF qw^8wgK VNK ppV#uupi)ax Parameters ========== gens : list A list of generators on which polycyclic subgroup is to be defined. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> S = SymmetricGroup(8) >>> G = S.sylow_subgroup(2) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> gens = [G[0], G[1]] >>> ipcgs = collector.induced_pcgs(gens) >>> [gen.order() for gen in ipcgs] [2, 2, 2] >>> G = S.sylow_subgroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> gens = [G[0], G[1]] >>> ipcgs = collector.induced_pcgs(gens) >>> [gen.order() for gen in ipcgs] [3] r<)r"r poprrrp)rr~rrwryrrrss&& r induced_pcgsCollector.induced_pcgss>CDII  aA 1 A 1 A3tyy>!CaxBsBwq!45!A# *ASSSA * +s C%C%c ^.\V4,pTpVPV4p\V4FwrgVPV4V8XgKVPV4VPV4,pWPV^, ,,pWx),V,pWV&VPV4pK V^8XdV#R#)z. Return the exponent vector for induced pcgs. F)r"rr2rr ) ripcgsryrFrrr8rsfs &&& rconstructive_membership_test&Collector.constructive_membership_testsCE N  JJqM&FA**S/Q&))!,T-B-B3-GG++AaC00"IaK!JJqM ' 6Hr)rr4r6r r r )NN)r%r&r'r(__doc__rrGrOrVrZrer5rrrrrrr+r,r-s@rrr*s]2:8t%2N2h$)NrjwrCJ TD6 +ZrrN) sympy.ntheory.primetestrsympy.combinatorics.perm_groupsrsympy.printing.defaultsrsympy.combinatorics.free_groupsrr rrjrrrs,+<36 o F\ \ r