CoCalc Public FilesPSET10-f.ipynbOpen with one click!
Author: Daishi Kiyohara
Views : 86
Description: Sage worksheet for problemset 10 problem 2
Compute Environment: Ubuntu 20.04 (Default)
In [1]:
A9=AlternatingGroup(9) for a in A9: for b in A9: if A9.subgroup([a,b]) == A9 and a**3==a and b**4==b: print (a,b) break
--------------------------------------------------------------------------- KeyboardInterrupt Traceback (most recent call last) <ipython-input-1-64887185b270> in <module> 2 for a in A9: 3 for b in A9: ----> 4 if A9.subgroup([a,b]) == A9 and a**Integer(3)==a and b**Integer(4)==b: 5 print (a,b) 6 break src/cysignals/signals.pyx in cysignals.signals.python_check_interrupt() KeyboardInterrupt:
In [4]:
A9=AlternatingGroup(9) for a,b in ((a_1,b_1) for a_1 in A9 for b_1 in A9): if a.ord()==2 and b.ord()==3: if A9.subgroup([a,b]) == A9: print (a,b) break
--------------------------------------------------------------------------- AttributeError Traceback (most recent call last) <ipython-input-4-a1008408c3e3> in <module> 1 A9=AlternatingGroup(Integer(9)) 2 for a,b in ((a_1,b_1) for a_1 in A9 for b_1 in A9): ----> 3 if a.ord()==Integer(2) and b.ord()==Integer(3): 4 if A9.subgroup([a,b]) == A9: 5 print (a,b) /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/structure/element.pyx in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4703)() 491 AttributeError: 'LeftZeroSemigroup_with_category.element_class' object has no attribute 'blah_blah' 492 """ --> 493 return self.getattr_from_category(name) 494 495 cdef getattr_from_category(self, name): /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/structure/element.pyx in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4815)() 504 else: 505 cls = P._abstract_element_class --> 506 return getattr_from_other_class(self, cls, name) 507 508 def __dir__(self): /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/cpython/getattr.pyx in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2620)() 370 dummy_error_message.cls = type(self) 371 dummy_error_message.name = name --> 372 raise AttributeError(dummy_error_message) 373 attribute = <object>attr 374 # Check for a descriptor (__get__ in Python) AttributeError: 'sage.groups.perm_gps.permgroup_element.PermutationGroupElement' object has no attribute 'ord'
In [7]:
A9=AlternatingGroup(9) for a in A9: if a.order()==2: for b in A9: if b.order()==3: if A9.subgroup([a,b]) == A9: print (a,b) break
(1,3)(2,7)(4,8)(5,9) (1,2,7)(3,8,5)(4,9,6) (1,9)(2,3)(5,7)(6,8) (1,5,3)(2,8,6)(4,9,7) (1,6)(2,7)(4,8)(5,9) (1,2,7)(3,8,5)(4,9,6) (1,3)(2,5)(4,8)(6,7) (1,2,9)(3,5,4)(6,7,8) (1,9)(2,3)(4,5)(6,8) (1,6,2)(3,5,4)(7,9,8) (1,7)(2,6)(3,4)(5,8) (1,7,6)(2,8,4)(3,9,5) (1,3)(2,8)(4,7)(6,9) (1,2,7)(3,8,5)(4,9,6) (1,6)(2,7)(3,9)(5,8) (1,9,5)(2,7,6)(3,8,4) (1,9)(2,4)(3,7)(6,8) (1,3,5)(2,9,7)(4,8,6) (1,6)(2,8)(3,4)(7,9) (1,3,5)(2,9,7)(4,8,6) (1,3)(2,5)(4,9)(6,7) (1,6,5)(2,8,7)(3,4,9) (1,3)(2,9)(4,6)(7,8) (1,6,5)(2,8,7)(3,4,9) (1,7)(2,6)(3,9)(5,8) (1,9,5)(2,7,6)(3,8,4) (1,3)(2,4)(6,7)(8,9) (1,9,5)(2,7,6)(3,8,4) (2,7)(3,6)(4,9)(5,8) (1,5,3)(2,8,6)(4,9,7) (1,3)(4,5)(6,7)(8,9) (1,3,5)(2,9,7)(4,8,6) (1,3)(2,5)(6,7)(8,9) (1,9,5)(2,7,6)(3,8,4) (1,3)(2,6)(4,8)(5,9) (1,6,2)(3,5,4)(7,9,8) (1,4)(2,7)(5,8)(6,9) (1,9,5)(2,7,6)(3,8,4)
--------------------------------------------------------------------------- KeyboardInterrupt Traceback (most recent call last) <ipython-input-7-1da3029ad08b> in <module> 4 for b in A9: 5 if b.order()==Integer(3): ----> 6 if A9.subgroup([a,b]) == A9: 7 print (a,b) 8 break /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/structure/richcmp.pyx in sage.structure.richcmp.slot_tp_richcompare (build/cythonized/sage/structure/richcmp.c:1878)() 271 Function to put in the ``tp_richcompare`` slot. 272 """ --> 273 return self.__richcmp__(other, op) 274 275 /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/groups/perm_gps/permgroup.py in __richcmp__(self, other, op) 4864 return NotImplemented 4865 -> 4866 c = PermutationGroup_generic.__richcmp__(self, other, op_EQ) 4867 if not c: 4868 return PermutationGroup_generic.__richcmp__(self, other, op) /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/groups/perm_gps/permgroup.py in __richcmp__(self, right, op) 713 gRight = right._libgap_() 714 if op in [op_EQ,op_NE]: --> 715 return gSelf._richcmp_(gRight, op) 716 717 if gSelf.IsSubgroup(gRight): /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/libs/gap/element.pyx in sage.libs.gap.element.GapElement._richcmp_ (build/cythonized/sage/libs/gap/element.c:9658)() 861 return hash(str(self)) 862 --> 863 cpdef _richcmp_(self, other, int op): 864 """ 865 Compare ``self`` with ``other``. /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/libs/gap/element.pyx in sage.libs.gap.element.GapElement._richcmp_ (build/cythonized/sage/libs/gap/element.c:9404)() 929 return self._compare_equal(other) or self._compare_less(other) 930 elif op == Py_EQ: --> 931 return self._compare_equal(other) 932 elif op == Py_GT: 933 return not self._compare_less(other) and not self._compare_equal(other) /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/libs/gap/element.pyx in sage.libs.gap.element.GapElement._compare_equal (build/cythonized/sage/libs/gap/element.c:9762)() 953 return id(self) == id(other) 954 cdef GapElement c_other = <GapElement>other --> 955 sig_on() 956 try: 957 GAP_Enter() KeyboardInterrupt:
In [11]:
A9=AlternatingGroup(9) for a in A9: if a.order()==2: for b in A9: if b.order()==3: if A9.subgroup([a,b]) == A9: H=A9.subgroup([a,b]).as_finitely_presented_group().simplified() G=SL(2,Integers()); S=G([0,-1,1,0]); T=G([1,1,0,1]) for i in range(2,len(H.relations())): print(H.relations()[i].subs(a=S,b=S*T),"\n") break
[-1 0] [14 -1] [ 1 0] [-15 1] [ 1 0] [-18 1] [ 1 0] [-20 1] [ 1 0] [-24 1] [-1 0] [24 -1] [ 1 0] [-24 1] [ 1 0] [-24 1] [-1 0] [30 -1] [-1 0] [38 -1] [ 1 0] [-38 1] [ 1 0] [-39 1] [ -1 -45] [ 0 -1] [ 1 48] [ 0 1] [ 1 51] [ 0 1] [-1 0] [15 -1] [ 1 24] [ 0 1] [ 1 0] [-24 1] [ 1 0] [-28 1] [ 1 0] [-28 1] [ 1 0] [-29 1] [ 1 0] [-33 1] [ -1 -38] [ 0 -1] [-1 0] [44 -1] [-1 0] [46 -1] [-1 0] [50 -1] [ 1 0] [-50 1] [-1 0] [15 -1] [ 1 0] [-24 1] [ 1 0] [-28 1] [ 1 0] [-28 1] [-1 0] [30 -1] [-1 0] [31 -1] [ 1 31] [ 0 1] [ 1 0] [-32 1] [-1 0] [38 -1] [-1 0] [38 -1] [-1 0] [38 -1] [-1 0] [40 -1] [ 1 0] [-48 1] [ 1 51] [ 0 1] [ 1 0] [-15 1] [ 1 0] [-18 1] [ 1 0] [-20 1] [ 1 0] [-24 1] [ 1 0] [-30 1] [ 1 0] [-30 1] [-1 0] [30 -1] [-1 0] [32 -1] [-1 0] [36 -1] [-1 0] [45 -1] [-1 0] [51 -1] [-1 0] [53 -1] [ 1 61] [ 0 1] [ 1 0] [-63 1] [-1 0] [63 -1] [-1 0] [96 -1] [ 1 0] [-15 1] [ 1 24] [ 0 1] [-1 0] [27 -1] [ 1 0] [-28 1] [ 1 28] [ 0 1] [ 1 0] [-28 1] [ 1 0] [-28 1] [ -1 -33] [ 0 -1] [ 1 0] [-35 1] [ -1 -36] [ 0 -1] [ 1 0] [-36 1] [-1 0] [39 -1] [ 1 46] [ 0 1] [ 1 0] [-48 1] [ 1 0] [15 1] [119508 53911] [ 50141 22619] [33588241 52102575] [56733915 88006486] [ 203471149 695803884] [ 531767916 1818470005] [ 384482134414 -169740266973] [-526511993957 232443274783] [-2738038845544 -1979646147183] [-2117508679473 -1530992851240] [10475987429429 17081945151359] [ 6202346160768 10113427267997] [191508235080089 521480005881363] [103387733264841 281526461399756] [ -569503398719896 -774961458460025] [ -928621371317687 -1263637361763056] [2096134342728547 868186982525820] [5056579650401663 2094358428796663] [-1615710887487374 -2598182141384301] [-2182163819651099 -3509080188603200] [281693410859735135 110403563141103494] [ 48804514404865261 19127860574457681] [ -754798588957958723 1061914081847140296] [ 985968808998884249 -1387143772994368835] [ 52273914397704629905 -40344481833764353854] [-11105586115180638496 8571179764125047857] [-124814744742810687423 -44352648775647958603] [ -45029043466232924558 -16000972911305420325] [59938792538616692458489 12560855077498245992023] [42377822790323601775859 8880754316600751374022] [-165515424969580647400611 -564807914517549949133225] [ -67640303791659044664628 -230817030672046282362791] [ 1197149893509438656269899 -1693224433232363401544344] [-5712645747347144614902956 8079849825198916970205635] [-18373445864139486777476362 -26127899446870546982615703] [-31365913862847754707687583 -44603796670890358796051425] [ 1 0] [-15 1] [ 1 0] [-18 1] [ 1 0] [-20 1] [ 1 24] [ 0 1] [ 1 24] [ 0 1] [ 1 28] [ 0 1] [-1 0] [30 -1] [-1 0] [31 -1] [ 1 0] [-31 1] [-1 0] [39 -1] [ 1 42] [ 0 1] [ -1 -43] [ 0 -1] [ 1 0] [-59 1] [119508 53911] [ 50141 22619] [ -28652104 -7763365] [-101781395 -27577944] [-768210187 -450031145] [-285474407 -167235968] [ -498939946 -689239207] [-1906884863 -2634184377] [-1377455834 -1962518153] [-3120103093 -4445339595] [-18753416456 -11405800987] [-14831696205 -9020616356] [-14975211813 -4390281500] [-50618446399 -14839805377] [70760351569 13083129945] [86963016925 16078897654] [377888654221 600623181828] [208263657964 331018090933] [519689014502 179035774557] [217599477937 74964238205] [ 454198788424 -1550273523735] [-1678985648505 5730722018524] [ -3274648482745 -12335803273277] [ -618342945117 -2329336100218] [142859544220579 42233773454388] [102626213146903 30339535661935] [-365710291403131 -83314436526438] [-828076396543227 -188648555976217] [1891855886353448405 2607196171611135953] [4285281269703483273 5905613107868883314] [-1 0] [14 -1] [ 1 0] [-15 1] [ 1 0] [-24 1] [ 1 0] [-24 1] [ 1 0] [-28 1] [ 1 0] [-28 1] [ 1 0] [-30 1] [-1 0] [32 -1] [ 1 0] [-34 1] [ -1 -36] [ 0 -1] [-1 0] [36 -1] [ 1 0] [-39 1] [ -1 -40] [ 0 -1] [ 1 0] [-40 1] [ 1 0] [-40 1] [ 1 0] [-15 1] [ 1 0] [-20 1] [ 1 0] [-26 1] [ 1 0] [-26 1] [-1 0] [26 -1] [ 1 32] [ 0 1] [ 1 0] [-32 1] [ 1 0] [-32 1] [ 1 0] [-33 1] [-1 0] [34 -1] [ 1 0] [-38 1] [ 1 0] [-40 1] [ 1 0] [-43 1] [-1 0] [46 -1] [-1 0] [51 -1] [ 1 0] [-58 1] [19371 10855] [15197 8516] [ 43601 -27040] [-113906 70641] [65597 53911] [93119 76530] [ 33487 -20748] [-159796 99007] [9189282 3265355] [3587449 1274778] [ 5855101 15959302] [ 2388424 6510149] [ 10107648 -36146407] [-12281897 43921835] [ -51930939 -14231263] [-167459087 -45890838] [1557482521 854361489] [ 572472813 314031598] [3529309795 4802600656] [5708333304 7767763915] [1191757004 5528798881] [ 729859291 3385963093] [ 2778933387 12893377174] [ 1717221305 7967366933] [-33445150586 7194006981] [ 54621164115 -11748939056] [110757848459 625647878790] [ 87503394214 494288339279] [-8662563485365 -5356528757966] [-5482208485179 -3389944265071] [3089591080357 5011240158039] [ 855227863150 1387158397543] [ -95754682995 247091743499] [ 456820700086 -1178810474017] [ -8682155347241 -4864221782993] [-11996684188505 -6721203459226] [203120133635988247057 77165499085523664367] [740284709777553686418 281234745531133427551] [ 1 0] [15 1] [1103 1426] [ 782 1011] [3070 5229] [2241 3817] [25359 92059] [14006 50845] [ 51874 22443] [141347 61153] [-110878 -46637] [ -87643 -36864] [ 48589 132618] [ 10295 28099] [1213808 947103] [3907633 3049025] [ 9178274 15038517] [23683413 38805053] [-7414755 -4553518] [-6432818 -3950495] [ 244441 1028280] [ 135300 569161] [ 17957829 31089715] [ 61968995 107284594] [40163759 47363680] [10883600 12834639] [122722897 54260455] [100875030 44600683] [ 23974252 114400247] [ 37867401 180695524] [ 651012663 1131433238] [ 277194208 481752135] [ 96511 155701] [ 28275 45616] [ 107337 -73513] [-140261 96062] [ 5031733 1292264] [17874317 4590533] [212350319 80723124] [172230444 65471903] [ 234334241 413619382] [ 603531592 1065283345] [-18753416456 -11405800987] [-14831696205 -9020616356] [-350777034 -91395275] [-985646491 -256811089] [-14975211813 -4390281500] [-50618446399 -14839805377] [377888654221 600623181828] [208263657964 331018090933] [-5730722018524 -1550273523735] [-1678985648505 -454198788424] [120334732494509 205628578706521] [ 73187518576039 125063189254480] [ -330220265171428 -758431267745689] [ -523900826521663 -1203265849918586] [-365710291403131 -83314436526438] [-828076396543227 -188648555976217] [ -870348302415853 2970826136481609] [ 3787814482207630 -12929235609066907] [ 8300294033183311 -3498613131310525] [-18548910168645691 7818453229256816] [ 84395688772766589 49964955551705081] [204248046901642006 120921396973730483] [ 60993630059966673849 21012646979452286051] [259847650563036404878 89518970167097861571] [ 1 0] [15 1] [ 2901 2000] [10300 7101] [ 43601 -27040] [-113906 70641] [119508 50141] [ 53911 22619] [-180135 -109564] [-142286 -86543] [295401 538265] [180883 329596] [1936616 3336435] [1141477 1966556] [ 834641 1025280] [1409760 1731761] [ 692206 -423943] [-1503605 920886] [-3450378 2439269] [ 523657 -370203] [42892881 30195610] [12668480 8918321] [ -3470585 8228778] [ 7750173 -18375707] [ 96162613 62461702] [228064322 148137465] [-2880208785 -1987984552] [-1754773607 -1211184009] [ -7762839698 -10979742383] [-21899128469 -30974076286] [ 135458817123 -75589784243] [-240223264567 134051257234] [-566368739356397 -215115230901461] [-234697229154146 -89141481747031] [2262784717636705 1459092625340131] [ 662953270246169 427486636276908] [ 1 0] [15 1] [65597 53911] [93119 76530] [ 57511 207844] [ 20748 74983] [ -7936617 -11491202] [-12836533 -18585651] [-43066064 -11650005] [-27763955 -7510559] [ 2707133 -9238170] [-11788142 40227377] [ -28652104 -7763365] [-101781395 -27577944] [-65351109870 -39746451257] [-51871740427 -31548318102] [ 74928070698 21966449951] [253267910821 74249835014] [-4051736370019 -5147811105330] [-1678985648505 -2133184436929] [ -86052924347 490208032251] [ 289257798264 -1647782421995] [-162158031001517 -88959075420045] [-114744880904774 -62948461148843] [122488901181931 46528259362732] [344979108895876 131042709164243] [ 4197406814380097 -1623242560393632] [-5174532694362354 2001121661798657] [2763142942423999 3351472833487422] [4470606858841155 5422490891283589] [119508 53911] [ 50141 22619] [ 95731 -20748] [-169624 36763] [ 8393958 12153373] [13576397 19656879] [ -28652104 -7763365] [-101781395 -27577944] [-1240964909 -1006513016] [-2865754614 -2324335925] [-18753416456 -11405800987] [-14831696205 -9020616356] [-14975211813 -4390281500] [-50618446399 -14839805377] [ -27783489050 -106184767181] [ -11459125629 -43795240577] [377888654221 600623181828] [208263657964 331018090933] [ 50284633493 116479845582] [ 21842221103 50595547079] [ -15887048212 54221102113] [ 69141504763 -235973891435] [ 454198788424 -1550273523735] [-1678985648505 5730722018524] [207253887903 878850653723] [127202447773 539396174960] [5959573337456 2274927811835] [7352453112445 2806627106271] [ -333631872868991 -1256811804235783] [ -62998796894495 -237320346251046] [ 16933581617178329 -10466120360526030] [-11796885764338215 7291288345265819] [-233 -377] [-377 -610] [ -1 -15] [ 0 -1] [1576 2241] [3735 5311] [78077 50024] [56446 36165] [ 32649 -24889] [ -9050 6899] [ 61153 22443] [141347 51874] [119508 53911] [ 50141 22619] [ 33487 -20748] [-159796 99007] [ -24949 -43135] [ -95015 -164274] [ 41491 49159] [136653 161908] [-172876 -486351] [ -53453 -150379] [-1206903 -1560320] [ -740830 -957767] [4352921 7025200] [1826552 2947881] [ -8154983 -1731786] [-13279521 -2820029] [ -55507619 -150310996] [ -23410563 -63394271] [-51568157 -13597172] [-94871257 -25015065] [270779912 195819865] [ 71735207 51876738] [-17180080 -7458563] [-54165733 -23515521] [ 124560166 -220672329] [ -33770973 59829073] [-4785913344105250 -1819868390249059] [-1713809634997161 -651685009189966] [ 96511 155701] [ 28275 45616] [ -28652104 -7763365] [-101781395 -27577944] [ 120929027 -198413576] [-170278202 279382939] [299839807 686557654] [125546750 287470443] [2152835803 2929165420] [1685641743 2293497487] [ -30433450613 -11616452002] [-102869570742 -39265328345] [-165501292019 -130344294769] [ -93410305795 -73567404124] [377888654221 600623181828] [208263657964 331018090933] [254009873356 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--------------------------------------------------------------------------- GAPError Traceback (most recent call last) <ipython-input-11-84a7232411b4> in <module> 8 G=SL(Integer(2),Integers()); S=G([Integer(0),-Integer(1),Integer(1),Integer(0)]); T=G([Integer(1),Integer(1),Integer(0),Integer(1)]) 9 for i in range(Integer(2),len(H.relations())): ---> 10 print(H.relations()[i].subs(a=S,b=S*T),"\n") 11 break /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/structure/element.pyx in sage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:12164)() 1511 cdef int cl = classify_elements(left, right) 1512 if HAVE_SAME_PARENT(cl): -> 1513 return (<Element>left)._mul_(right) 1514 if BOTH_ARE_ELEMENT(cl): 1515 return coercion_model.bin_op(left, right, mul) /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/groups/libgap_wrapper.pyx in sage.groups.libgap_wrapper.ElementLibGAP._mul_ (build/cythonized/sage/groups/libgap_wrapper.c:7092)() 628 """ 629 P = (<ElementLibGAP> left)._parent --> 630 return P.element_class(P, (<ElementLibGAP> left)._libgap * \ 631 (<ElementLibGAP> right)._libgap) 632 /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/structure/element.pyx in sage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:12164)() 1511 cdef int cl = classify_elements(left, right) 1512 if HAVE_SAME_PARENT(cl): -> 1513 return (<Element>left)._mul_(right) 1514 if BOTH_ARE_ELEMENT(cl): 1515 return coercion_model.bin_op(left, right, mul) /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/libs/gap/element.pyx in sage.libs.gap.element.GapElement._mul_ (build/cythonized/sage/libs/gap/element.c:10749)() 1063 try: 1064 sig_GAP_Enter() -> 1065 sig_on() 1066 result = PROD(self.value, (<GapElement>right).value) 1067 sig_off() GAPError: Error, no method found! Error, no 1st choice method found for `CallFuncList' on 2 arguments
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