ó ~9­\c@ sçddlmZmZddlZddlmZmZmZddlm Z ddl m Z ddl m Z ddlmZddlmZd efd „ƒYZged „Zd „Zd „Zd„Zed„Zd„ZdS(iÿÿÿÿ(tprint_functiontdivisionN(tFpGroupt FpSubgrouptsimplify_presentation(t FreeGroup(tPermutationGroup(tigcd(ttotient(tStGroupHomomorphismcB s•eZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z d „Z d „Zd „Zd„Zd„ZRS(sÖ A class representing group homomorphisms. Instantiate using `homomorphism()`. References ========== .. [1] Holt, D., Eick, B. and O'Brien, E. (2005). Handbook of computational group theory. cC s:||_||_||_d|_d|_d|_dS(N(tdomaintcodomaintimagestNonet _inversest_kernelt_image(tselfR R R ((s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pyt__init__s      c C sG|jƒ}i}xLt|jjƒƒD]5}|j|}||kpM|js(|||ès (Rt is_subgroupR RER R R (RRTR ((RTRs@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pytcomposeÞsc sdt|tƒ s#|jˆjƒ r2tdƒ‚n|}‡fd†|jDƒ}t|ˆj|ƒS(sh Return the restriction of the homomorphism to the subgroup `H` of the domain. s'Given H is not a subgroup of the domainc si|]}ˆ|ƒ|“qS(((RSR!(R(s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pys ôs (RRRUR RERR R (RtHR R ((Rs@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pyt restrict_toës #cC sØ|j|jƒƒs$tdƒ‚ng}t|jƒjƒ}x’|jD]‡}|j|ƒ}||kr†|j|ƒt|ƒ}nxG|jƒjD]6}|||kr–|j||ƒt|ƒ}q–q–WqIW|S(s„ Return the subgroup of the domain that is the inverse image of the subgroup `H` of the homomorphism image s&Given H is not a subgroup of the image( RURRERRRR.R:R2(RRWR tPthth_iR((s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pytinvert_subgroup÷s  (t__name__t __module__t__doc__RR%R.R2R1RRDRLRMRPRQRRRVRXR\(((s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pyR s    # *      cC s¤t|tttfƒs'tdƒ‚nt|tttfƒsNtdƒ‚n|j}tg|D]}||k^qaƒr‹tdƒ‚ntg|D]}||k^q•ƒr¿tdƒ‚n|rìt|ƒt|ƒkrìtdƒ‚nt |ƒ}t |ƒ}|j |j gt|ƒt|ƒƒ|j g|D]}||kr8|^q8ƒt t ||ƒƒ}|r”t|||ƒ r”tdƒ‚nt|||ƒS(sy 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`, don't check whether the given images actually define a homomorphism. sThe domain must be a groupsThe codomain must be a groupsCThe supplied generators must be a subset of the domain's generatorss+The images must be elements of the codomains>The number of images must be equal to the number of generatorss-The given images do not define a homomorphism(RRRRt TypeErrorRtanyRER-RtextendRtdicttzipt_check_homomorphismR (R R R R tcheckRR!((s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pyt homomorphism s& %%  ',c stˆdƒrˆj}nˆjƒj‰ˆjƒj}|j‰‡‡‡‡fd†}x¦|D]ž}t|tƒræ|j||ƒˆƒ}|dkrõ|j ƒ}|j||ƒˆƒ}|dkrã| rãt dƒ‚qãqõn||ƒj }|sat SqaWt S(Ntrelatorsc sÞ|jr ˆSˆ}|j}d}d}x«|t|ƒkrÕ||d}tˆtƒrwˆjˆj||ƒ}n ||}|ˆkr¢|ˆ||}n|ˆ|d|}|t|ƒ7}|d7}q+W|SdS(Niiiÿÿÿÿ(RRFR-RRRtindexRG(R>R"tr_arrR/tjtpowerR$(R R RR (s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pyR=s"     sÖCan't determine if the images define a homomorphism. Try increasing the maximum number of rewriting rules (group._rewriting_system.set_max(new_value); the current value is stored in group._rewriting_system.maxeqns)(thasattrRht presentationRRRRtequalsRtmake_confluentt RuntimeErrorRtFalseR8(R R R trelsRR>R$tsuccess((R R RR s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pyRe5s$     c sÕddlm‰ddlm}|tˆƒƒ}|j‰tˆƒ‰‡‡‡fd†|jDƒ}|jdˆƒt |||ƒ}t|j ƒtˆƒkr¼|j tˆƒ|_ nt |jgƒ|_ |S(s‘ Return the homomorphism induced by the action of the permutation group `group` on the set `omega` that is closed under the action. iÿÿÿÿ(R&(tSymmetricGroupc  sCi|]9}ˆˆgˆD]}ˆj||Aƒ^qƒ|“qS((Ri(RSR!to(R&Rtomega(s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pys |s tbase( R(R&t sympy.combinatorics.named_groupsRuR-RRRt_schreier_simsR tbasic_stabilizersRR(tgroupRwRuR R RW((R&RRws@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pytorbit_homomorphismqs  c  s ddlm‰ddlm}t|ƒ}d}g‰dg|‰xHt|ƒD]:}|||krRˆj|ƒ|ˆ|<|d7}qRqRWx&t|ƒD]}ˆ||ˆ|i|]4}ˆgˆD]}ˆˆ||A^qƒ|“qS(((RSR!R/(R&tbRRK(s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pys ¥s N( R(R&RyRuR-RR,R:RR ( R|tblocksRutntmR/R R RW((R&R~RRKs@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pytblock_homomorphism…s$       "c C s½t|ttfƒs$tdƒ‚nt|ttfƒsHtdƒ‚nt|tƒrÚt|tƒrÚt|ƒ}t|ƒ}|j|jkrÚ|jjƒ|jjƒkrÚ|s¸tStt |||j|jƒfSn|}|j ƒ}|j ƒ}|t j krt dƒ‚nt|tƒrX|t j krCt dƒ‚n|jƒ\}}n||ksv|j|jkrŠ|s€tStdfS|s¸|}t|t|ƒƒdkr¸tSnt|jƒ}xßtj|t|ƒƒD]Å} t| ƒ} | j|jgt|jƒt| ƒƒtt|| ƒƒ} t||| ƒràt|tƒrd|j| ƒ} nt |||j| dtƒ} | jƒr¥|s˜tSt| fSqàqàW|s³tStdfS(sÏ 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 (boolean) -- 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 Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.free_groups import free_group >>> from sympy.combinatorics.fp_groups import FpGroup >>> from sympy.combinatorics.homomorphisms import homomorphism, 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. s2The group must be a PermutationGroup or an FpGroups<Isomorphism methods are not implemented for infinite groups.iRfN(RRRR`RRRhtsortR8RgR5R R6R7t_to_perm_groupt is_abelianRrRRRRt itertoolst permutationsR-RbRRcRdReR.RQ( R;RWt isomorphismt_Htg_orderth_ordert h_isomorphismR€R tsubsetR t_imagestT((s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pytgroup_isomorphism©sX5  0"    * cC st||dtƒS(s  Check if the groups are isomorphic to each other Parameters ========== G (a finite `FpGroup` or a `PermutationGroup`) -- First group H (a finite `FpGroup` or a `PermutationGroup`) -- Second group Returns ======= boolean Rˆ(RRr(R;RW((s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pyt is_isomorphics(t __future__RRR†tsympy.combinatorics.fp_groupsRRRR)Rtsympy.combinatorics.perm_groupsRtsympy.core.numbersRtsympy.ntheory.factor_RR4R tobjectR R8RgReR}R‚RR‘(((s@lib/python2.7/site-packages/sympy/combinatorics/homomorphisms.pyts ÿ) <  $ q