\hS`SSKJr SSKJr SSKJr SSKJr "SS\5r"SS\5r g ) isprime)PermutationGroup)DefaultPrinting) free_groupc2\rSrSrSrSrSSjrSrSrSr g) PolycyclicGroupTNc|XlX lX0lU(d[URX#5UlgUUlg)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 rrs U/var/www/auris/envauris/lib/python3.13/site-packages/sympy/combinatorics/pc_groups.py__init__PolycyclicGroup.__init__ s/$ ",PY499iH_hc:[SUR55$)Nc38# UHn[U5v M g7fNr).0orders r 1PolycyclicGroup.is_prime_order..$sC/Be75>>/Bs)allrrs ris_prime_orderPolycyclicGroup.is_prime_order#sCt/B/BCCCrc,[UR5$r)lenr rs rlengthPolycyclicGroup.length&s499~r)rr r rr) __name__ __module__ __qualname____firstlineno__is_group is_solvablerr r$__static_attributes__rrr r sHKi.Drr cj\rSrSrSrSSjrSrSrSrSr S r S r S r S r S rSrSrSrSrg)r*z References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" Section 8.1.3 Nc8XlX lX0lU(d&[SR [ U555SOUUl[ URR5VVs0sHupgXv_M snnUlUR5Ul gs snnf)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:{}rN) r r rrformatr# enumeratesymbolsindex pc_relatorspc_presentation)rr r r free_group_r6iss rrCollector.__init__5sy, ",IT*V]]3t9%=>qAZe'01H1H'IJ'Itqad'IJ #//1Ks)Bc~U(dgURnURnURn[[ U55H0nX%upgX4U(dMUS:dXsXFS- :dM,Xg44s $ [[ U5S- 5H.nX%upgX%S-upXFXH:dMU S:aSOSn Xg4X44s $ g)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_formrr4ranger#) rwordarrayrer4r8s1e1s2e2es rminimal_uncollected_subword%Collector.minimal_uncollected_subwordRsF   s5z"AXFB)}}"q&BEIq,@|# # s5z!|$AXFBQ3ZFBy59$aAR2'** %rc0n0nURR5H(up4[UR5S:XaXAU'M$XBU'M* X4$)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<)r6itemsr#r>)rpower_relatorsconjugate_relatorskeyvalues r relationsCollector.relationssV<..446JC3>>"a'&+s#*/3' 7 11rcSnSn[[U5[U5- S-5H8nURXU[U5-5U:XdM&UnU[U5-n X44$ X44$)a| Returns the start and ending index of a given subword in a word. Parameters ========== word : FreeGroupElement word defined on free group elements for a polycyclic group. w : FreeGroupElement subword of a given word, whose starting and ending index to be computed. Returns ======= (i, j) A tuple containing starting and ending index of ``w`` in the given word. If not exists, (-1,-1) is returned. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics import free_group >>> G = SymmetricGroup(4) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> F, x1, x2 = free_group("x1, x2") >>> word = x2**2*x1**7 >>> w = x2**2*x1 >>> collector.subword_index(word, w) (0, 3) >>> w = x1**7 >>> collector.subword_index(word, w) (2, 9) >>> w = x1**8 >>> collector.subword_index(word, w) (-1, -1) r=r<)r?r#subword)rr@wlowhighr8s r subword_indexCollector.subword_indexspVs4yQ')*A||AQx(A-Qxy + yrcURnUSSnUSSnUS4US4US44nURRU5nURU$)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 rr<r=)r>rdtyper6)rrTrArCrErNs r map_relationCollector.map_relationsd>  1Xa[ 1Xa[Bx"a2q'*oo##C(##C((rcURnURU5nU(dU$URXR"U55upEUS:XaMGUSupg[ U5S:XaUR UR UnXx-n XyU-- n USSU44n UR"U 5n URU (aCURU Rn U SupUSSU 4XU-44nUR"U5nO&U S:waUSSU 44nUR"U5nOSnURUR"U5U5n[ U5S:XaxUSSS:alUSunnUS44nUR"U5nURUR"U55nUX--nUR"U5nURXEU5nO[ U5S:XazUSSS:anUSunnUS44nUR"U5nURUR"U55nUS-X--nUR"U5nURXEU5nGMR)a Return the collected form of a word. Explanation =========== A word ``w`` is called collected, if `w = {x_{i_1}}^{a_1} * \ldots * {x_{i_r}}^{a_r}` with `i_1 < i_2< \ldots < i_r` and `a_j` is in `\{1, \ldots, {s_j}-1\}`. Otherwise w is uncollected. Parameters ========== word : FreeGroupElement An uncollected word. Returns ======= word A collected word of form `w = {x_{i_1}}^{a_1}, \ldots, {x_{i_r}}^{a_r}` with `i_1, i_2, \ldots, i_r` and `a_j \in \{1, \ldots, {s_j}-1\}`. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics import free_group >>> G = SymmetricGroup(4) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> F, x0, x1, x2, x3 = free_group("x0, x1, x2, x3") >>> word = x3*x2*x1*x0 >>> collected_word = collector.collected_word(word) >>> free_to_perm = {} >>> free_group = collector.free_group >>> for sym, gen in zip(free_group.symbols, collector.pcgs): ... free_to_perm[sym] = gen >>> G1 = PermutationGroup() >>> for w in word: ... sym = w[0] ... perm = free_to_perm[sym] ... G1 = PermutationGroup([perm] + G1.generators) >>> G2 = PermutationGroup() >>> for w in collected_word: ... sym = w[0] ... perm = free_to_perm[sym] ... G2 = PermutationGroup([perm] + G2.generators) The two are not identical, but they are equivalent: >>> G1.equals(G2), G1 == G2 (True, False) See Also ======== minimal_uncollected_subword r=rr<N) rrHrWrZr#rr4r6r>eliminate_wordr[substituted_word)rr@rrTrUrVrCrDrBqrrN presentationsymexpword_rErFs rcollected_wordCollector.collected_wordsB__ 006AZ W**41A1A!1DEICbyqTFB1v{((B8HtG!Q}' &&s+'',#'#7#7#<#G#GL+AHCd1gq\C3<8E&,,U3EAv"#A$q'1 0 * 0 0 7 $**:+;+;A+>F1v{qtAw{1B1g[%%b)))**:*:1*=>59 "((/,,S>Q11a11B1g[%%b)))**:*:1*=>Buy("((/,,S>]rcURnURn0n0nURn[XQR5HupgUS-XFS-'XtU'M USSS2nUR SSS2nUSSS2n/n [ U5GHMupX*n XFU -n Xn U RXk-SS9nUR5 URnUH nXU-nM URU5nU(aUOSX<'X0l U RU5 [U 5S:dMU [U 5S- nUUn[[U 5S- 5HnXIUnUS-U-U-n US-U U-U-nU RUSS9nUR5 URnUH nXU-nM URU5nU(aUOSX<'X0l M GMP U$)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 r=NToriginalr-r<)rrr zip generatorsr r2generator_productreverseidentityrgr6appendr#r?)rr rel_orderr5 perm_to_freer genr9seriescollected_gensr8rBrelationGlr@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+&JrcURn[5nURHn[U/UR-5nM UR USS9nUR 5 0n[ URUR5HuptUS-XdS-'XvU'M URnUH nXU-nM URU5n URn S/[U5-n U Rn U Hn U SXU S'M U $)a: Return the exponent vector of length equal to the length of polycyclic generating sequence. Explanation =========== For a given generator/element ``g`` of the polycyclic group, it can be represented as `g = {x_1}^{e_1}, \ldots, {x_n}^{e_n}`, where `x_i` represents polycyclic generators and ``n`` is the number of generators in the free_group equal to the length of pcgs. Parameters ========== element : Permutation Generator of a polycyclic group. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> from sympy.combinatorics.permutations import Permutation >>> G = SymmetricGroup(4) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> pcgs = PcGroup.pcgs >>> collector.exponent_vector(G[0]) [1, 0, 0, 0] >>> exp = collector.exponent_vector(G[1]) >>> g = Permutation() >>> for i in range(len(exp)): ... g = g*pcgs[i]**exp[i] if exp[i] else g >>> assert g == G[1] References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" Section 8.1.1, Definition 8.4 Trjr=rr<) rrr rmrnrorlrprgr4r#r>) relementrrxrzrrsrdrTr@r4 exp_vectorts rexponent_vectorCollector.exponent_vectors Z__  A !q||!34A""7t"<  *//;FC"%r'LB !O<   Aq/!A""1% SZ( A&'dJQqT{ #rcURU5n[S[U55[UR5S-5$)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 c3B# UHupU(dMUS-v M g7f)r<Nr-)rr8xs rr"Collector.depth..as@%:TQaSQqS%:s  r<)rnextr2r#r )rrrs rdepthCollector.depthAs:>))'2 @Yz%:@#dii.QRBRSSrcURU5nURU5nU[UR5S-:waX#S- $g)a Return the leading non-zero exponent. Explanation =========== The leading exponent for a given element `g` is defined by `\mathrm{leading\_exponent}[g]` `= e_i`, if `\mathrm{depth}[g] = i`. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> G = SymmetricGroup(3) >>> PcGroup = G.polycyclic_group() >>> collector = PcGroup.collector >>> collector.leading_exponent(G[1]) 1 r<N)rrr#r )rrrrs rleading_exponentCollector.leading_exponentcsG*))'2  7# C N1$ $Ag& &rctUnURU5nU[UR5:aXS- S:waXS- nURU5URU5S--nX`RUS- -nXV*-U-nURU5nU[UR5:a XS- S:waMU$)Nr<r=)rr#r rr)rzrzhdkrGs r_siftCollector._sift~s  JJqM#dii. QsVq[A#A%%a($*?*?*BR)GGA''!,,A2aA 1 A #dii. QsVq[ rcS/[UR5-nUnU(aURS5nURX$5nUR U5nU[UR5:a8UH+nUS:wdM UR US-US--U-U-5 M- XRUS- 'U(aMUVs/sH owS:wdM UPM nnU$s snf)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<rr=)r#r poprrrq)rrrrxrzrrrts r induced_pcgsCollector.induced_pcgss>CDII  aA 1 A 1 A3tyy>!CaxBsBwq!45!A#a *ASSA * +s 4 C C cS/[U5-nUnURU5n[U5HupgURU5U:XdMURU5URU5-nXRUS- -nXx*-U-nXU'URU5nURU5U:XaMkM US:XaU$g)z. Return the exponent vector for induced pcgs. rr<F)r#rr2rr) ripcgsrzrGrrr8rtfs rconstructive_membership_test&Collector.constructive_membership_testsCE N  JJqM&FA**S/Q&))!,T-B-B3-GG++AaC00"IaK!JJqM **S/Q&' 6Hr)rr4r6r r r)NN)r&r'r(r)__doc__rrHrPrWr[rgr5rrrrrrr,r-rrrr*sU2:8t%2N2h$)NrjwrCJ TD6 +ZrrN) sympy.ntheory.primetestrsympy.combinatorics.perm_groupsrsympy.printing.defaultsrsympy.combinatorics.free_groupsrr rr-rrrs,+<36 o F\ \ r