+ jI^RIt^RIHtHtHt^RIHt^RIHt^RI H t ^RI H t ^RI Ht!RR 4tRR ltR tR tR tRRltRtR#)N)FpGroup FpSubgroupsimplify_presentation) FreeGroup)PermutationGroup)igcd)totient)Sc~a]tRt^ toRtRtRtRtRtRt Rt Rt R t R t R tR tR tRtRtRtRtVtR#)GroupHomomorphismz A class representing group homomorphisms. Instantiate using `homomorphism()`. References ========== .. [1] Holt, D., Eick, B. and O'Brien, E. (2005). Handbook of computational group theory. cTWnW nW0nRVnRVnRVnR#N)domaincodomainimages _inverses_kernel_image)selfrrrs&&&&b/home/luca/screen-checking/.venv/lib/python3.14/site-packages/sympy/combinatorics/homomorphisms.py__init__GroupHomomorphism.__init__s&     c VP4p/p\VPP44F6pVPV,pWB9dKVP'dK2W2V&K8 \ VP \4'dVPpM VPpVFpWb9gVP'dKVPPp\ VP \4'dVPV,RRR1,pMTpVF7p W9dWrV ,,pKWrV R,,R,,pK9 WrV&K V#)z Return a dictionary with `{gen: inverse}` where `gen` is a rewriting generator of `codomain` (e.g. strong generator for permutation groups) and `inverse` is an element of its preimage N) imagelistrkeys is_identity isinstancerr strong_gens generatorsridentity_strong_gens_slp) rrinverseskvgensgwpartsss & r_invsGroupHomomorphism._invss dkk&&()A AAM}}} * dmm%5 6 6$$D##DA}  $$A$--)9::..q1$B$7=1+ A1b5/2--A  QKrc V^RIHp^RIHp\ WV34'EdN\ VP \ 4'dVP PV4pVPfVP4VnVP4pVPPp\ VP \4'dVPV4RRR1,pMTp\\!V44FqpWg,pVP"'dKWP9dWPPV,,pKJWPPVR,,R,,pKs V#\ V\$4'd!VU u.uFqP'V 4NK up #R#uup i)a Return an element of the preimage of ``g`` or of each element of ``g`` if ``g`` is a list. Explanation =========== If the codomain is an FpGroup, the inverse for equal elements might not always be the same unless the FpGroup's rewriting system is confluent. However, making a system confluent can be time-consuming. If it's important, try `self.codomain.make_confluent()` first.  Permutation)FreeGroupElementNr)sympy.combinatoricsr1sympy.combinatorics.free_groupsr2r rrreducerr-rrr#rgenerator_productrangelenrrinvert) rr)r1r2rr*r(ir,es && rr9GroupHomomorphism.invert?s5 4D a'78 9 9$--11MM((+~~%!%JJLE $$A$--)9::..q1$B$73t9%G===&..++A..B/33A&H 4 ,-.AqKKNA. .!.sF&c `VPfVP4VnVP#)z Compute the kernel of `self`. )r_compute_kernelrs&rkernelGroupHomomorphism.kernelms( << //1DL||rcdVPpVP4pV\PJd \ R4h.p\ V\ 4'd\ VP4pM \WRR7pVP4P4pVP4V,V8wdVP4pW`PV!V44R,,pWt9gKXVPV4\ V\ 4'd\ V4pK\WRR7pKV#)z9Kernel computation is not implemented for infinite groupsT)normalr) rorderr InfinityNotImplementedErrorr rr#rrrandomr9append)rGG_orderr(Kr:rr&s& rr>!GroupHomomorphism._compute_kernelvs KK'') ajj %KM M a) * * ,A140A JJL   ggikW$ A++d1g&**Az Aa!122(.A"148Arc \VPf\\VPP 444p\ VP \4'd-VP PV4VnVP#\VP V4VnVP#)z Compute the image of `self`. ) rrsetrvaluesr rrsubgroupr)rrPs& rrGroupHomomorphism.images| ;; #dkk00234F$--)9::"mm44V< {{)? {{rc WP9dH\V\\34'd!VUu.uFq P V4NK up#\ R4hVP 'dVPP#VPpVPPp\VP\4'dgVPPVRR7pVFApW`P9dW6,V,pK$W6R,,R,V,pKC V#^pVPFLwrV ^8dW,R,pMW,pWCV,V ,,pV\V 4, pKN V#uupi)z Apply `self` to `elem`. z2The supplied element does not belong to the domainT)originalr)rr rtuple_apply ValueErrorrrr#rrr6 array_formabs) relemr;rvaluer(r)r:_ps && rrVGroupHomomorphism._applys9 {{ "$u ..0451 A55QR R    ==)) )[[FMM**E$++'788{{44TD4IAKK' & % &"u r 1% 7   OODA1u GRK G!)Q,.EQKA , /6sFc$VPV4#r)rV)rrZs&&r__call__GroupHomomorphism.__call__s{{4  rc DVP4P4^8H#)z) Check if the homomorphism is injective )r@rDr?s&r is_injectiveGroupHomomorphism.is_injectives {{}""$))rc VP4P4pVPP4pV\PJdV\PJdR#W8H#)z* Check if the homomorphism is surjective N)rrDrr rE)rimoths& r is_surjectiveGroupHomomorphism.is_surjectivesL ZZ\   !mm!!#  qzz 19 rc RVP4;'dVP4#)z% Check if `self` is an isomorphism. )rcrhr?s&ris_isomorphism GroupHomomorphism.is_isomorphisms$   ";;t'9'9';;rc DVP4P4^8H#)z[ Check is `self` is a trivial homomorphism, i.e. all elements are mapped to the identity. )rrDr?s&r is_trivialGroupHomomorphism.is_trivials zz|!!#q((rc  VP4PVP4'g \R4hVPUu/uFq"V!V!V44bK pp\ VPVP V4#uupi)z Return the composition of `self` and `other`, i.e. the homomorphism phi such that for all g in the domain of `other`, phi(g) = self(other(g)) z?The image of `other` must be a subgroup of the domain of `self`)r is_subgrouprrWrr r)rotherr)rs&& rcomposeGroupHomomorphism.composesp{{}((55+, ,-2\\:\T%(^#\: t}}fEE;s Bc \V\4'd"VPVP4'g \ R4hTpVP Uu/uF q3V!V4bK pp\ W PV4#uupi)zP Return the restriction of the homomorphism to the subgroup `H` of the domain. z'Given H is not a subgroup of the domain)r rrqrrWr"r r)rHrr)rs&& r restrict_toGroupHomomorphism.restrict_tosi !-..ammDKK6P6PFG G&'ll3lT!W*l3 ??4sA<c VPVP44'g \R4h.p\VP4P4pVP FpVP V4pWS9dVPV4\V4pVP4P F4pWe,V9gKVPWe,4\V4pK6 K V#)zn Return the subgroup of the domain that is the inverse image of the subgroup ``H`` of the homomorphism image z&Given H is not a subgroup of the image) rqrrWrr#r"r9rHr@)rrvr(Phh_ir&s&& rinvert_subgroup!GroupHomomorphism.invert_subgroups }}TZZ\**EF F TZZ\22 3A++a.C| C $T*[[]--5>KK&(.A. r)rrrrrrN)__name__ __module__ __qualname____firstlineno____doc__rr-r9r@r>rrVr`rcrhrkrnrsrwr}__static_attributes____classdictcell__) __classdict__s@rr r s_!F,/\. @!* <) F @rr caa\V\\\34'g \ R4h\S\\\34'g \ R4hVP o\ ;QJdV3RlV4F 'dK RM RM!V3RlV44'g \R4h\ ;QJdV3RlV4F 'dK RM RM!V3RlV44'g \R4hV'd%\V4\V48wd \R 4h\V4p\V4pVPSP.\S4\V4, ,4TPSUu.uF qUV9gK VNK up4\\W#44pV'd\VSV4'g \R 4h\VSV4#uupi) a^ Create (if possible) a group homomorphism from the group ``domain`` to the group ``codomain`` defined by the images of the domain's generators ``gens``. ``gens`` and ``images`` can be either lists or tuples of equal sizes. If ``gens`` is a proper subset of the group's generators, the unspecified generators will be mapped to the identity. If the images are not specified, a trivial homomorphism will be created. If the given images of the generators do not define a homomorphism, an exception is raised. If ``check`` is ``False``, do not check whether the given images actually define a homomorphism. zThe domain must be a groupzThe codomain must be a groupc3,<"TF qS9xK R#5ir).0r)r"s& r homomorphism..#s-1JFTzCThe supplied generators must be a subset of the domain's generatorsc3,<"TF qS9xK R#5irr)rr)rs& rrr%s-fH}frz+The images must be elements of the codomainz>The number of images must be equal to the number of generatorsz-The given images do not define a homomorphism)r rrr TypeErrorr"allrWr8rextendr#dictzip_check_homomorphismr )rrr(rcheckr)r"s&f&&& @r homomorphismr se f/)D E E455 h!17I F G G677""J 3--333-- - -^__ 3-f-333-f- - -FGG #f+T*YZZ :D &\F MM8$$%s:s6{'BCDKKJ8Jq4-J89 #d" #F (6BBHII VXv 66 9s G+G+cdaa a \VR4'dTMVP4pVPpVPpVUu.uFqfP^,NK pp\ \ WpP44o VPo V VV 3RlpVFp \V\4'd]VPV!V 4S 4p V f@VP4p VPV!V 4S 4p V fV 'g \R4hMV!V 4Pp V 'dKR# R#uupi)a9 Check that a given mapping of generators to images defines a homomorphism. Parameters ========== domain : PermutationGroup, FpGroup, FreeGroup codomain : PermutationGroup, FpGroup, FreeGroup images : dict The set of keys must be equal to domain.generators. The values must be elements of the codomain. relatorscv<SpVPF%wr#SV,pVSV,V,,pK' V#r)rX)rLr*symbolpowerr)r#rsymbols_to_domain_generatorss& rr#_check_homomorphism.._imageJs< \\MF,V4A E! !A*rzCan't determine if the images define a homomorphism. Try increasing the maximum number of rewriting rules (group._rewriting_system.set_max(new_value); the current value is stored in group._rewriting_system.maxeqns)FT)hasattr presentationrr"ext_reprrr#r requalsmake_confluent RuntimeErrorr)rrrpresrelsr(r)symbolsrrLr,successr#rs&&f @@rrr6sVZ006f6I6I6KD ==D ??D%)*Tyy||TG*#'G5F5F(G#H   H h ( (q 84Ay#113OOF1Ix89W&(+,,q %%Aq'( ?+sD-c^RIHp^RIHpV!\ V44pVP p\ V4pVPUUu/uF5qfYR!VUu.uFqqPWv, 4NK up4,bK7 pppVPVR7\WV4p \ VP4\ V48d$VP\ V4,V n V #\VP .4V n V #uupiuuppi)z Return the homomorphism induced by the action of the permutation group ``group`` on the set ``omega`` that is closed under the action. r0SymmetricGroup)base)r3r1 sympy.combinatorics.named_groupsrr8r#rr"index_schreier_simsr basic_stabilizersrr) groupomegar1rrr#r)orrvs && rorbit_homomorphismrgs 0?c%j)H  H KERWRbRb cRbQ+&GA{{13'7&GHHHRbF c e$%62A 5 " "#c%j0++CJ7  H%enn%56 H'H cs DD.DDc ^RIHp^RIHp\ V4p^p.pR.V,p\ V4F0pW,V8XgKVP V4WWV&V^, pK2 \ V4FpWqV,,Wx&K V!V4p \ V4p VPU Uu/uF.qT!V Uu.uFqWh,V , ,NK up4bK0 p p p\W V 4p V #uupiuupp i)aJ Return the homomorphism induced by the action of the permutation group ``group`` on the block system ``blocks``. The latter should be of the same form as returned by the ``minimal_block`` method for permutation groups, namely a list of length ``group.degree`` where the i-th entry is a representative of the block i belongs to. r0rN) r3r1rrr8r7rHr"r )rblocksr1rnmr]br:rr#r)rrvs&& rblock_homomorphismr{s0? F A A A qA 1X 9> HHQKaD FA  1X|a HQxHEJEUEU VEU:AQii:;;EUF V%62A H; Vs* C45C/ C4/C4c &\V\\34'g \R4h\V\\34'g \R4h\V\4'd\V\4'd\ V4p\ V4pVP VP 8XdeVP P4VP P48Xd.V'gR#R\WVP VP 43#TpVP4pVP4pV\PJd \R4h\V\4'd2V\PJd \R4hVP4wr6WE8wgVPVP8wd V'gR#R#V'gTp\V\!V44^8XdR#\#VP 4p\$P&!V\)V44Fp \#V 4p V P+VP,.\)VP 4\)V 4, ,4\/\1W44p \3WV 4'gK|\V\4'dXP5V 4p \WVP V RR7p V P74'gKV'gR#RV 3u# V'gR#R#)a Compute an isomorphism between 2 given groups. Parameters ========== G : A finite ``FpGroup`` or a ``PermutationGroup``. First group. H : A finite ``FpGroup`` or a ``PermutationGroup`` Second group. isomorphism : bool This is used to avoid the computation of homomorphism when the user only wants to check if there exists an isomorphism between the groups. Returns ======= If isomorphism = False -- Returns a boolean. If isomorphism = True -- Returns a boolean and an isomorphism between `G` and `H`. Examples ======== >>> from sympy.combinatorics import free_group, Permutation >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.fp_groups import FpGroup >>> from sympy.combinatorics.homomorphisms import group_isomorphism >>> from sympy.combinatorics.named_groups import DihedralGroup, AlternatingGroup >>> D = DihedralGroup(8) >>> p = Permutation(0, 1, 2, 3, 4, 5, 6, 7) >>> P = PermutationGroup(p) >>> group_isomorphism(D, P) (False, None) >>> F, a, b = free_group("a, b") >>> G = FpGroup(F, [a**3, b**3, (a*b)**2]) >>> H = AlternatingGroup(4) >>> (check, T) = group_isomorphism(G, H) >>> check True >>> T(b*a*b**-1*a**-1*b**-1) (0 2 3) Notes ===== Uses the approach suggested by Robert Tarjan to compute the isomorphism between two groups. First, the generators of ``G`` are mapped to the elements of ``H`` and we check if the mapping induces an isomorphism. z2The group must be a PermutationGroup or an FpGroupTzrsMTT5<#9"BBH'7R/b (" Hrh6r