+ jRt^RIHt^RIHt^RIHtHtHt^RI H t ^RI H t H t Ht^RIHt^RIHt^RIHt^R IHt^R IHt^R IHt^R IHt^R IHt]R%Rl4t]R%Rl4t ]R4t!Rt"!RR]4t#!RR]4t$R%Rlt%Rt&Rt'R%Rlt(RR/Rlt)Rt*Rt+R&Rlt,R t-R!t.R"t/R'R$lt0]t1R##)(z.Finitely Presented Groups and its algorithms. )S)symbols) FreeGroupFreeGroupElement free_group)RewritingSystem) CosetTablecoset_enumeration_rcoset_enumeration_c)PermutationGroup)invariant_factors)Matrix)gcd)DefaultPrinting)public)pollute)productcR\W4pV3\VP4,#N)FpGrouptuple _generatorsfr_grprelators _fp_groups&& ^/home/luca/screen-checking/.venv/lib/python3.14/site-packages/sympy/combinatorics/fp_groups.pyfp_grouprs$)I <% 5 56 66c2\W4pW"P3#r)rrrs&& r xfp_groupr s)I ,, --rc\\V4p\VPUu.uFq3PNK upVP4V#uupir)rrrname generators)fr_grpmrrsyms&& r vfp_groupr&!sB*I !2!2 3!2#XX!2 3Y5I5IJ  4sA cV#)zParse the passed relators.)relss&r_parse_relatorsr*(s KrcTa]tRt^2toRtRtRtRtRtRt Rt Rt Rt ] R 4tR tR(R ltR)R ltRtR)RltR*RltRtR+RltRtRtR*RltRt]tRtRtRtRtRt Rt!Rt"Rt#Rt$] R 4t%] R!4t&] R"4t'] R#4t(] R$4t)R%t*R&t+R't,Vt-R #),rz The FpGroup would take a FreeGroup and a list/tuple of relators, the relators would be specified in such a way that each of them be equal to the identity of the provided free group. TFc\V4pWnW nVP4Vn\ R\ 3RV/4VnRVnRVn RVn RVn \V4Vn RVnR#)FpGroupElementgroupNF)r*rrrr#typer-dtype _coset_table_is_standardized_order_centerr_rewriting_system_perm_isomorphism)selfrrs&&&r__init__FpGroup.__init__=sx"8,  **,*^,=O !!&  !0!6!%rc.VPP#r)rr#r7s&rrFpGroup._generatorsQs)))rc :VPP4R#)z5 Try to make the group's rewriting system confluent N)r5make_confluentr;s&rr>FpGroup.make_confluentTs --/rc 8VPPV4#)z Return the reduced form of `word` in `self` according to the group's rewriting system. If it's confluent, the reduced form is the unique normal form of the word in the group. )r5reduce)r7words&&rrAFpGroup.reduce\s%%,,T22rc VPWR,,4VP8XdR#VPP'dR#R#)z Compare `word1` and `word2` for equality in the group using the group's rewriting system. If the system is confluent, the returned answer is necessarily correct. (If it is not, `False` could be returned in some cases where in fact `word1 == word2`) TFN)rAidentityr5 is_confluent)r7word1word2s&&&requalsFpGroup.equalses: ;;uBY '4== 8  # # 0 0 0rc.VPP#r)rrFr;s&rrFFpGroup.identityts'''rcWP9#r)r)r7gs&&r __contains__FpGroup.__contains__xsOO##rNc  Ja\;QJdRV4F 'dK RM RM !RV44'g \R4h\;QJdV3RlV4F 'dK RM RM!V3RlV44'g \R4hV'd\SWRR7wrEpM\SWR7wrEV'd\V^,PV4pM\\ R 4^,.4pV'd^R IHpWC!VSVPXRR 73#V#) a! Return the subgroup generated by `gens` using the Reidemeister-Schreier algorithm homomorphism -- When set to True, return a dictionary containing the images of the presentation generators in the original group. Examples ======== >>> from sympy.combinatorics.fp_groups import FpGroup >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x**3, y**5, (x*y)**2]) >>> H = [x*y, x**-1*y**-1*x*y*x] >>> K, T = f.subgroup(H, homomorphism=True) >>> T(K.generators) [x*y, x**-1*y**2*x**-1] c3B"TFp\V\4xK R#5ir) isinstancer).0rOs& r #FpGroup.subgroup..sADq:a!122DsFTz&Generators must be `FreeGroupElement`sc3T<"TFqPSP8HxK R#5ir)r.r)rUrOr7s& rrVrWs>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x, y**2]) >>> f.order(strategy="coset_table_based") 2 r[)r3r1lenr{rrorderr#absr array_form _is_infiniterInfinity_finite_index_subgrouprfindex)r7rsrrdrYinds&& rr FpGroup.ordersa ;; ";;     (d//556DK{{ 1 $////1DK{{ !Q &ct}}"M}!<<?1#5#5}"MNODK{{   **DK{{ 113GD!''l!---":"@"@"BB {{#jjn {{#Ns"G c  \4pVPF"pVPVP44K$ \VP4V8:gR#.pVPF<pTP VPUu.uFqTP V4NK up4K> \\V44p^\V49dR#R#uupi)zY Test if the group is infinite. Return `True` if the test succeeds and `None` otherwise TN) setrupdatecontains_generatorsr#append exponent_sumr r )r7 used_gensr abelian_relsrelrOms& rrFpGroup._is_infinites E A   Q224 54??#y0 ==C   doo No!1!1!!4o N O! 6,' ( !!$ $ !OsC c <VP4p\VP4pVPVP4V'g\ VP4^8Xd.V.VPUu.uF qDV8wgK VNK up,pMVP Pp^pWS9g!VR,V9gVP'd#V^ 8dVP4pV^, pKHW%.VPUu.uF qDV8wgK VNK up,p\ V4^,^,pVRVpWRp Rp Rp ^p Rp V 'gV ^, \P8d\V \P4p VPWV RR7p V P4'dT p TpM.VPWV RR7p V P4'dT p T pV 'dKV ^,p KV 'gR#V P4XV 3#uupiuupi)z Find the elements of `self` that generate a finite index subgroup and, if found, return the list of elements and the coset table of `self` by the subgroup, otherwise return `(None, None)` NTrjrE)NN)most_frequent_generatorlistr#extendrrrrF is_identityrandomrrnminrtrorp)r7sgenr)rOrandimidhalf1half2draft1draft2rrYhalfs&& rrFpGroup._finite_index_subgroups**,DOO$ DMM"4??#q(ED18QQDD// bD(8D.Hs84a&&q))4s )r#rsumrmax)r7rdr)rOfreqss& ` rrFpGroup.most_frequent_generatorEsP}}BFG$Q8488$GKKE +,,HsAc^RIpVPPp\VP ^^44F<pW!P VP 4VP ^R.4,,pK> V#)NrE)rrrFrangerandintchoicer#)r7rrrs& rrFpGroup.randomKsZ OO $ $v~~a*+A--0&--22GGGA,rc |V.8XdVP4#VPW4p\VP4#)a Return the index of subgroup ``H`` in group ``self``. Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x**5, y**4, y*x*y**3*x**3]) >>> f.index([x]) 4 )rrtrr{)r7rrrsrYs&&& rr FpGroup.indexRs5$ 7::< &&q3Aqww< rcVPP^8d RVPP,pV#R\VP4,pV#)zz)rrankstrr#r7str_forms& r__str__FpGroup.__str__jsI ??  " $69M9MMH93t;OOHrc  J^RIHp^RIHpVP 4\ P Jd \R4hVP'd VPpVP4pWC3#VP.4pVPpVUUu.uFJp\\V44Uu.uF(qV,^VPV4,,NK* upNKL p ppV Uu.uF q!V4NK p p\V 4pV!WWiRR7pW0nWC3#uupiuuppiuupi)z Return an isomorphic permutation group and the isomorphism. The implementation is dependent on coset enumeration so will only terminate for finite groups. ) Permutationr]z>Permutation presentation of infinite groups is not implementedFr^)sympy.combinatoricsrrcrZrrrNotImplementedErrorr6imager|r#rrrr ) r7rrZTPrYrdrOrimagess & r_to_perm_groupFpGroup._to_perm_groupws 4B ::<1:: %%'NO O  ! ! !&&A At   $A??DNRSdeCFmDmtAdjjmO,,mDdFS.45fk!nfF5 (ATd%@A%& "t ES5s D(.D D$D Dc VP4wr4\W14!V!pRp\V\4'dV.Rre.pVF/pVPp VP VP V 44K1 V'd V^,#T#)a  Given the name of a `PermutationGroup` method (returning a subgroup or a list of subgroups) and (optionally) additional arguments it takes, return a list or a list of lists containing the generators of this (or these) subgroups in terms of the generators of `self`. FT)rgetattrrTr r#rinvert) r7 method_nameargsrr perm_resultsingleresultr.rds &&* r_perm_group_listFpGroup._perm_group_lists""$a-t4  k#3 4 4#.- E##D MM!((4. )!#vay..rc $VPR4#)zg Return the list of lists containing the generators of the subgroups in the derived series of `self`. derived_seriesrr;s&rrFpGroup.derived_seriess $$%566rc $VPR4#)zm Return the list of lists containing the generators of the subgroups in the lower central series of `self`. lower_central_seriesrr;s&rrFpGroup.lower_central_seriess $$%;<22rc $VPR4#)z Check if `self` is solvable. is_solvablerr;s&rrFpGroup.is_solvables ""=11rc \VP4wrVPVP4#)z List the elements of `self`. )rrelementsr7rrs& rrFpGroup.elementss' ""$xx ##rc \VP4^8:dR#VP4wrVP# \d \R4hi;i)z& Return ``True`` if group is Cyclic. Tz2Check for infinite Cyclic group is not implemented)rr#rr is_cyclicrs& rrFpGroup.is_cyclics\ t 1 $ <&&(DA{{# <%';< < rArJpropertyrFrPrfrtrxr|rrrrrrr__repr__rrrrrrrrrrrrrrrr__static_attributes____classdictcell__) __classdict__s@rrr2s/ HJ(*3 (($$L8%,!F,.`-  0H 2/&7=/9;> 113322$$   & & &rrcVaa]tRtRtoRtR V3RlltRtRtRtRt ] t Rt Vt V;t #) FpSubgroupi'z The class implementing a subgroup of an FpGroup or a FreeGroup (only finite index subgroups are supported at this point). This is to be used if one wishes to check if an element of the original group belongs to the subgroup c<\SV`4Wn\VUu0uFqDVP8wgKVkK up4VnRVnRVnW0nR#uupir) superr8parentrrFr# _min_wordsrYnormal)r7GrdrrO __class__s&&&& rr8FpSubgroup.__init__/sQ  4C4a ?4CD  Ds AAc$aaaaa\SP\4'EdSPEf|Rp.pSPF<pSP 'dVP 4pVPV!V44K> VEFpVEFpWV8Xg%\V\4'gVR,V8XdK0\V\4'd(V^,^,V^,^,R,rM!V^,V\V4^, ,r\V\4'd(V^,^,V^,^,R,rM!V^,V\V4^, ,rYVr\V\4'd-V^,V^,,V^,R,,p \V\4'd-V^,V^,,V^,R,,p V R,V8Xd>W,P'g&V!W,4p W9dVPV 4V R,V8XgEKW,P'dEKV!W,4p W9gEKVPV 4EK EK VSnSPoVV3Rlo/oVVV3RloSP 'dVP 4pS!V4#SPf-SPPSP4pVSn ^pSPp\\V44F5pVPV,VPVV,,,pK7 V^8H#)NcjVPRR7wrVP'gW!3.#WR,.#)TremovedrE)cyclic_reductionr)wprs& r_process)FpSubgroup.__contains__.._processAs7--d-;DA===!"x !b5z)rcl<VPRR7wrVP'gSP'dVS9#SUu.uF4p\V\4'gKV^,V8XgK+V^,NK6 ppVUu.uFq P V4'gKVNK up.8g#uupiuupi)Tr )r"rrrTrpower_of)r#rrt min_wordsr7s& r _is_subword,FpSubgroup.__contains__.._is_subwords))$)7===DKKK >)'0Oy!Jq%4HDEaDAI1yAO'(:q!JJqMAAq:b@@O:sB, B,0 B,B1 B1c&<\V4^8XdR#^pV\V48dlV^, pVP^V4pS!V4'gK;VPV\V44pVS9d S!V4SV&SV,'gKyR#R#)rTF)rsubword)r#rprefixrestr+ _word_breakknowns& rr1,FpSubgroup.__contains__.._word_breaksq6Q;#a&jFAYYq!_F&v.. 99QA/D5(&1$&7d T{{#rrE)rTrrrr#rr"rrrrrYrtrr{A_dict)r7rOr%rdr#w1w2s1s2r1r2p1p2newrYrjr+r1r2r*sf& @@@@rrPFpSubgroup.__contains__7s dkk9 - -& *A{{{..0KK ,) B"8Jr5,A,A<>FbL$ &b%00%'U1Xr!uQx|%'UBs2wqyM%b%00%'U1Xr!uQx|%'UBs2wqyM"$B%b%00!#Ar!uRUBY!6B%b%00!#Ar!uRUBY!6Br6R<0C0C0C"*25/C" $ C 0r6R<0C0C0C"*25/C" $ C 0M#R#'I AE ${{{&&(q> !vv~KK11$//BAA3q6]GGAJqxx!~.#6MrcVP'g\P#\VP\ 4'd\P #VPf,VPPVP4pWnVPP4\VPP4, #r) r#rOnerTrrrrYrtrrr{)r7rYs& rrFpSubgroup.orders|55L dkk9 - -::  66> --doo>AF{{  "3tvv||#444rcB\VP\4'dV\\ VP 44Uu.uF pRV,NK pp\ RPV44^,#VPPVPR7#uupi)zx_%d, r[) rTrrrrr#rjoinrfrY)r7rgen_symss& r to_FpGroupFpSubgroup.to_FpGroupsx dkk9 - -,1#doo2F,GH,Gq,GHHdii1215 5{{##dff#--IsBc\VP4^8dR\VP4,pV#R\VP4,pV#)rz z")rr#rrs& rrFpSubgroup.__str__sG t " $9CRRHr)rYrr#rrF)rrr r r r8rPrrGrrrr __classcell__)rrs@@rrr's.wr 5. HHrrc T\V.4pVPp^pVUu0uFp\V4V8gKVkK pp\V4V, Uu0uFqfP 4kK ppVP V4p .p \ WWP^,WqV4V #uupiuupi)a Implements the Low Index Subgroups algorithm, i.e find all subgroups of ``G`` upto a given index ``N``. This implements the method described in [Sim94]. This procedure involves a backtrack search over incomplete Coset Tables, rather than over forced coincidences. Parameters ========== G: An FpGroup < X|R > N: positive integer, representing the maximum index value for subgroups Y: (an optional argument) specifying a list of subgroup generators, such that each of the resulting subgroup contains the subgroup generated by Y. Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, low_index_subgroups >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x**2, y**3, (x*y)**4]) >>> L = low_index_subgroups(f, 4) >>> for coset_table in L: ... print(coset_table.table) [[0, 0, 0, 0]] [[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 3, 3]] [[0, 0, 1, 2], [2, 2, 2, 0], [1, 1, 0, 1]] [[1, 1, 0, 0], [0, 0, 1, 1]] References ========== .. [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" Section 5.4 .. [2] Marston Conder and Peter Dobcsanyi "Applications and Adaptions of the Low Index Subgroups Procedure" )rrrridentity_cyclic_reduction conjugatesdescendant_subgroupsA) rNYrYR len_short_relrR2R1 R1_c_listrs &&& rlow_index_subgroupsrYsR 1bA AM 7#c#h6##B 769Vb[ A[c ' ' )[B A R I Ay##a&"; H 8 BsB B B%c VPpVPpVP4'dM\WAP4FwrVP W4'dKR# VP !V4R#\\\VP44VP4F+wrVPV ,Ws,,eK)YrM VPVP.,p V FPpW8gK WP8Xg&VPV,VX ,,eK@\WW$VX X W4 KR R#r) r4 A_dict_invroromega scan_checkrrrr{rQntry_descendant)rrYrXxrVrRrSr4r[r#alphaundefined_coset undefined_genreachbetas&&&&&&& rrPrP s XXFJ}} GG,HA<<))-   c!''l 3QSS9HEwwu~fi(016 :133%Dx33;!''$- =0I"J"R"1?)44rc VP4p WyP8XdXWt8dRV PPR.\ V P 4,4V P PV4WyPV,V PV,&WYPV,V PV,&V PPWV34V PW)PV,,W)PV,,4'gR#VFp V P^V 4'dKR# \W4'd\W W&W4V4R#R#)zN Solves the problem of trying out each individual possibility for `\alpha^x. N)copyr^r{rrrQr$r4r[deduction_stackprocess_deductions_checkr]first_in_classrP) rrYrXrVrRrar`rerSDr#s &&&&&&&&& rr_r_%s A ss{tx vc!##h'  4"&GGEN188A;%*GGDM!,,q/"eZ( % %i &< ll1o & ( ( ||Aq!! aQ9:rcVPpRpR.V,pR.V,pRp\^V4EFTp\V^,4FpRWEV,&K VF7p VPV,VPV ,,V8wgK5RpM V'dRpKqWu^&^WG&^p\V4FpVPFp VPV,VPV ,,p VPWX,,VPV ,,p V eV fRpMBWL,fV^, pW4V &WV&WL,V 8dR#WL,V 8gKRpM V'gKRpEKR EKW R#)aw Checks whether the subgroup ``H=G1`` corresponding to the Coset Table could possibly be the canonical representative of its conjugacy class. Parameters ========== C: CosetTable Returns ======= bool: True/False If this returns False, then no descendant of C can have that property, and so we can abandon C. If it returns True, then we need to process further the node of the search tree corresponding to C, and so we call ``descendant_subgroups`` recursively on C. Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, CosetTable, first_in_class >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x**2, y**3, (x*y)**4]) >>> C = CosetTable(f, []) >>> C.table = [[0, 0, None, None]] >>> first_in_class(C) True >>> C.table = [[1, 1, 1, None], [0, 0, None, 1]]; C.p = [0, 1] >>> first_in_class(C) True >>> C.table = [[1, 1, 2, 1], [0, 0, 0, None], [None, None, None, 0]] >>> C.p = [0, 1, 2] >>> first_in_class(C) False >>> C.table = [[1, 1, 1, 2], [0, 0, 2, 0], [2, None, 0, 1]] >>> first_in_class(C) False # TODO:: Sims points out in [Sim94] that performance can be improved by # remembering some of the information computed by ``first_in_class``. If # the ``continue alpha`` statement is executed at line 14, then the same thing # will happen for that value of alpha in any descendant of the table C, and so # the values the values of alpha for which this occurs could profitably be # stored and passed through to the descendants of C. Of course this would # make the code more complicated. # The code below is taken directly from the function on page 208 of [Sim94] # nu[alpha] NFTrE)r^rr{r4rQ) rYrSr^lamdanumu next_alphararer#r`gammadeltas && rrjrj<skl A E B BJq!%'NDB$xL# Awwu~ahhqk*e3!   J 1 !HDSS ahhqk2)!((1+6=EM!%J9$QJE %uI %uI9u$ 9u$!%J'(z" //^ r change_gensFcl\V4^8Xd\V^,\4'g \R4hV^,p\ VP VP VR7wr4V'd\V^,PV4#\\.4.4#\V4^8XdIV^,R,V^,R,rCV'gW43#V^,PPpM \V4^8XdRpMRp\V4h.p.p\V4\V48XgCTpV'd,\V4\V48XgTp\W4V4wr4K3\V4pK[V'dVU u.uFqP^,^,NK p p \V 4^,p V PpV P p\!\#W44p \%V4F:wrVPpTpVFwppVV V,V,,pK VWM&K< W43#uup i)a For an instance of `FpGroup`, return a simplified isomorphic copy of the group (e.g. remove redundant generators or relators). Alternatively, a list of generators and relators can be passed in which case the simplified lists will be returned. By default, the generators of the group are unchanged. If you would like to remove redundant generators, set the keyword argument `change_gens = True`. z+The argument must be an instance of FpGrouprsNNNzNot enough argumentszToo many arguments)rrTr TypeErrorsimplify_presentationr#rr.rrF RuntimeErrorrelimination_technique_1_simplify_relatorsrrdictzip enumerate)rsrrrdr)rFr prev_gens prev_relsrOsymsFsubsr>rarr%r$s$* rrxrxs 4yA~$q'7++IJ J G*1<<:EG  47==$/ /y}b)) Ta!WQZad: 7==)) t9>&A$A1oII)nD ) #i.CI"=I0XFJD$!$',01Dq Q""D1 t Q ::||CO$dODA ACQ$s)Q,&DG $ :2s"H1cVR,pV'g.#V^,PPp/p\\V44FpW,pVP 4^8XgK"V^,p\ VP ^,^,4pVP ^,^,^8dW,R,W&VR,pWR9d+\WbV,P ^,^,4pWV,W%&K \VP44p\V4Fwr4WG9dK VPVRR7pVP4FpWR9gK W%,P ^,^,pV^,^,pVPWX,WXV, ,RR7pVPWX),WXV, ),RR7pK W@V&K VU u.uFqP4NK pp W, p\\V44pVP!4VP#V4V#uup i \$dT#i;i)a Simplifies a set of relators. All relators are checked to see if they are of the form `gen^n`. If any such relators are found then all other relators are processed for strings in the `gen` known order. Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import _simplify_relators >>> F, x, y = free_group("x, y") >>> w1 = [x**2*y**4, x**3] >>> _simplify_relators(w1) [x**3, x**-1*y**4] >>> w2 = [x**2*y**-4*x**5, x**3, x**2*y**8, y**5] >>> _simplify_relators(w2) [x**-1*y**-2, x**-1*y*x**-1, x**3, y**5] >>> w3 = [x**6*y**4, x**4] >>> _simplify_relators(w3) [x**4, x**2*y**4] >>> w4 = [x**2, x**5, y**3] >>> _simplify_relators(w4) [x, y**3] rvT_allrE)r.rFrrnumber_syllablesrrrrvaluesr~eliminate_wordsreliminate_wordrNrsortremovera) r)rFexpsrrrOexpone_syllables_wordsmax_exprs & rr{r{s: 7D  Aw}}%%H D 3t9 g    !Q &AAcnnQ'*+C~~a #a''2+rEy#Aw11!4Q78fDGt{{}-D/  % !!"5d!C((*Ayg((+A.7Q,((q3;7GPT(U((XckN8KTX(Y + Q"48 84a ' ' )4D 8D D ?DIIK  H K 9   K s;II I)(I)cNaVR,pVP4VR,p/p.p\4pVEFDpVP4o\;QJdV3RlV4F 'gK RM RM!V3RlV44'dKV\ S4oSPRR7SFpVP V4^8XgKWu9gK#VP V4pVPWx,4p VPV ^,\V44p VP^V 4p W,p V RV,,W7&VPV P44VPV4EKB EKG VU u.uF qV9gK V NK pp VU u.uF#qPVRR7P4NK% pp \ \V44pVPV4VUu.uF qV9gK VNK ppW3#uup iuup i \dL3i;iuupi)rvc3,<"TF qS9xK R#5irr()rUrOcontained_genss& rrV*elimination_technique_1..Ks;NqN"NsTFreverserrE)rrranyrrrrr.rrrrrNrra)rdr)rFredundant_gensredundant_relsrrrk gen_indexbkfwchirrOrs&&& @rrzrz<s 7D IIK 7DNNI002 3;N;333;N; ; ; n-d+!C""3'1,1E$$S)IIcf- [[QC9[[I.e&)BqDk#  !8!8!:;%%c*"& 7t!6AAtD 7`d e`d[\  nT  : T T V`dD e D ?D  H 7t!6AAtD 7 : 8 e    7s0HH)H H/H"<H" HHcF.p^pVPpVPpV'd/p/pVPV^&\VP4Uu.uF pR.\ VP 4,NK" upVn\VPVP 4EFUwrVPV ,VPV ,,p W8XdvRVPV ,VPV ,&RVPV ,VPV ,&V^, pV'dXV ,V ,W{&KKW9gKVPV ,VPV ,,eKV : RV : 2p VPV 4WPV ,VPV ,&V'gEK.XV ,V ,W{,R,,XV &EKX \\RP!V444p V P#^4VnWnV'dXVn\\\ VP44\\ VP 444EFwrVPV,V,R8Xd-VP$PVPV,V&KQ\+VPV,V,\,4'gKVP&VP/VPV,V,4,pWPV,V&VPV,V,p VR,VPV ,V^,&EK R#uupi)z Parameters ========== C -- Coset table. homomorphism -- When set to True, return a dictionary containing the images of the presentation generators in the original group. Nz _rDrE)rr#rFrr^rrQrrr\r{r4r[rrrrEpop_schreier_free_group_schreier_generators_schreier_gen_elemrTrr)rYrZyrqfXretaurrar`re y_alpha_xgrp_gensr>rs&& rdefine_schreier_generatorsrjsz A E A AA$)!##J /JqD6#acc(??J /ACAGGQSS)wwu~ahhqk* =&2ACCJqxx{ #)5ACCIall1o & QJEJqL  VE 188A;/7#$e,I HHY &/CCJqxx{ #|#&u:a< 2 #=i *Jtyy|,-H%\\!_A%$c!##hs133x9 33q6!9 $..77ACCF1I Aq 3 ' '&&qwwqss1vay'9:ACCF1I771:a=D "uACCIa!e :+ 0s&Nc aVPPpVUUu.uF+p\VP4Fp\ WV4NK K- pppVUu0uFp\ V4^8XgKVkK upo\ \V3RlV44p\\ V44F"pWE,pVPSRR7pWdV&K$ VPUu.uFpVS9dK VR,S9dKVNK upVn ^pV\ V48dYWE,pV^,pV\ V48d-VPWG,4'dWGK1V^, pK<V^, pKhW@n R#uuppiuupiuupi)c<VS9#rr()r order_1_genss&r'reidemeister_relators..s 3l#:rTrNrE) rrrr^rewriterrfilterrris_cyclic_conjugate_reidemeister_relators) rYrTrBcosetr)rr#r>rs & @rreidemeister_relatorsrs[ A01 JuQSSzeGAd #z #D J#3t!s1v{AAt3L :DA BD3t9  G  l  6Q *+)?)?I)?A- 12B,1F a)?IA A c$i- G E#d)m$$TW--GQ Q#9 K3Is#1E4E:,E: E?$ E?4E?c0VPPp\\V44FgpW$,pW0PV,VP V,,,pVP V,VP V,,pKi V#)a\ Parameters ========== C: CosetTable alpha: A live coset w: A word in `A*` Returns ======= rho(tau(alpha), w) Examples ======== >>> from sympy.combinatorics.fp_groups import FpGroup, CosetTable, define_schreier_generators, rewrite >>> from sympy.combinatorics import free_group >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x**2, y**3, (x*y)**6]) >>> C = CosetTable(f, []) >>> C.table = [[1, 1, 2, 3], [0, 0, 4, 5], [4, 4, 3, 0], [5, 5, 0, 2], [2, 2, 5, 1], [3, 3, 1, 4]] >>> C.p = [0, 1, 2, 3, 4, 5] >>> define_schreier_generators(C) >>> rewrite(C, 0, (x*y)**6) x_4*y_2*x_3*x_1*x_2*y_4*x_5 )rrFrrrr4r{)rYrar#vrx_is&&& rrrsl: ''A 3q6]d cc%j#' 'qxx}- Hrc VPpVPRR7VPp\\ V4^, RR4FpW,p\\ V4^, RR4FpW%,pVP V4^8XgK#VP V4pVPWg,4pVPV^,\ V44p VP^V4p W,RV,,p WW%VUu.uFqDPWk4NK ppK K WnW nVPVP3#uupi)a This technique eliminates one generator at a time. Heuristically this seems superior in that we may select for elimination the generator with shortest equivalent string at each stage. >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_r, reidemeister_relators, define_schreier_generators, elimination_technique_2 >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x**3, y**5, (x*y)**2]); H = [x*y, x**-1*y**-1*x*y*x] >>> C = coset_enumeration_r(f, H) >>> C.compress(); C.standardize() >>> define_schreier_generators(C) >>> reidemeister_relators(C) >>> elimination_technique_2(C) ([y_1, y_2], [y_2**-3, y_2*y_1*y_2*y_1*y_2*y_1, y_1**2]) TrrE) rrrrrrrrr.r) rYr)rdrrr>rrrrrrep_bys & relimination_technique_2rs+& # #DIIdI ! !D 3t9q="b )gs4y1}b"-A'C""3'1,$$S)IIcf- [[QC9[[I.%2a4GGCGH4C**374H.* $! ! !1#;#; ;; Is ENcV'g \W4pVP4VP4\W#R7\ V4VP VP rT\WERR7wrE\V4Vn \V4VnV'dDVUu.uFqbP\V4,NK ppVPVPV3#VPVP3#uupi)a Parameters ========== fp_group: A finitely presented group, an instance of FpGroup H: A subgroup whose presentation is to be found, given as a list of words in generators of `fp_grp` homomorphism: When set to True, return a homomorphism from the subgroup to the parent group Examples ======== >>> from sympy.combinatorics import free_group >>> from sympy.combinatorics.fp_groups import FpGroup, reidemeister_presentation >>> F, x, y = free_group("x, y") Example 5.6 Pg. 177 from [1] >>> f = FpGroup(F, [x**3, y**5, (x*y)**2]) >>> H = [x*y, x**-1*y**-1*x*y*x] >>> reidemeister_presentation(f, H) ((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1)) Example 5.8 Pg. 183 from [1] >>> f = FpGroup(F, [x**3, y**3, (x*y)**3]) >>> H = [x*y, x*y**-1] >>> reidemeister_presentation(f, H) ((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0)) Exercises Q2. Pg 187 from [1] >>> f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3]) >>> H = [x] >>> reidemeister_presentation(f, H) ((x_0,), (x_0**4,)) Example 5.9 Pg. 183 from [1] >>> f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2]) >>> H = [x] >>> reidemeister_presentation(f, H) ((x_0,), (x_0**6,)) r]Tru) r rprwrrrrrxrschreier_generatorsrr)fp_grprrrYrZrdr)rres&&&& rrbrb sV  *JJL!--/q<!'')A)A$&ttDJD!$KA#DkA;?@4C%%c#h//4@$$a&=&=uDD !"9"9 99As$C4)r(rKr)2r sympy.core.singletonrsympy.core.symbolrsympy.combinatorics.free_groupsrrr#sympy.combinatorics.rewritingsystemrsympy.combinatorics.coset_tablerr r rr sympy.matrices.normalformsr sympy.matricesr sympy.polys.polytoolsrsympy.printing.defaultsrsympy.utilitiesrsympy.utilities.magicr itertoolsrrr r&r*rrrYrPr_rjrxr{rzrrrrrbr-r(rrrs4"%<<?BB18!%3")77..  r&or&jbbR6 r48;.qn6U6pNb'\1%f$B" J&