问题
For a finite abelian group G, say,
G = AbelianGroup((4,4,5)),
I want Sage to return the automorphism group of G. Is this implemented?
回答1:
You can get it partway easily.
G = AbelianGroup((4,4,5))
gap(G).AutomorphismGroup()
Group( [ Pcgs([ f1, f2, f3, f4, f5 ]) -> [ f1*f3*f4, f2*f4, f3*f4, f4, f5 ],
Pcgs([ f1, f2, f3, f4, f5 ]) -> [ f1*f3*f4, f2*f4, f3*f4, f4, f5 ],
Pcgs([ f1, f2, f3, f4, f5 ]) -> [ f1, f2, f1*f2*f3, f2*f4, f5 ],
Pcgs([ f1, f2, f3, f4, f5 ]) -> [ f1, f2, f2*f3*f4, f4, f5 ],
Pcgs([ f1, f2, f3, f4, f5 ]) -> [ f1, f2, f3, f4, f5^2 ] ] )
Unfortunately this particular group type doesn't seem to have an obvious way to get it back into Sage. This answer luckily gives it: otherwise it's buried as a parameter in the official permutation group doc. Here you go, though the generators may not be what you were looking for:
G = AbelianGroup((4,4,5))
H = gap(G).AutomorphismGroup()
PermutationGroup(gap_group = H.AsPermGroup())
Permutation Group with generators [(1,21)(2,22)(3,24)(4,23)(5,17)(6,18)(7,20)(8,19)(9,13)(10,14)(11,16)(12,15)(25,45)(26,46)(27,48)(28,47)(29,41)(30,42)(31,44)(32,43)(33,37)(34,38)(35,40)(36,39)(49,81)(50,82)(51,84)(52,83)(53,77)(54,78)(55,80)(56,79)(57,73)(58,74)(59,76)(60,75)(61,93)(62,94)(63,96)(64,95)(65,89)(66,90)(67,92)(68,91)(69,85)(70,86)(71,88)(72,87)(97,117)(98,118)(99,120)(100,119)(101,113)(102,114)(103,116)(104,115)(105,109)(106,110)(107,112)(108,111)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)(127,140)(128,139)(129,133)(130,134)(131,136)(132,135)(145,177)(146,178)(147,180)(148,179)(149,173)(150,174)(151,176)(152,175)(153,169)(154,170)(155,172)(156,171)(157,189)(158,190)(159,192)(160,191)(161,185)(162,186)(163,188)(164,187)(165,181)(166,182)(167,184)(168,183)(193,213)(194,214)(195,216)(196,215)(197,209)(198,210)(199,212)(200,211)(201,205)(202,206)(203,208)(204,207)(217,237)(218,238)(219,240)(220,239)(221,233)(222,234)(223,236)(224,235)(225,229)(226,230)(227,232)(228,231)(241,273)(242,274)(243,276)(244,275)(245,269)(246,270)(247,272)(248,271)(249,265)(250,266)(251,268)(252,267)(253,285)(254,286)(255,288)(256,287)(257,281)(258,282)(259,284)(260,283)(261,277)(262,278)(263,280)(264,279)(289,309)(290,310)(291,312)(292,311)(293,305)(294,306)(295,308)(296,307)(297,301)(298,302)(299,304)(300,303)(313,333)(314,334)(315,336)(316,335)(317,329)(318,330)(319,332)(320,331)(321,325)(322,326)(323,328)(324,327)(337,369)(338,370)(339,372)(340,371)(341,365)(342,366)(343,368)(344,367)(345,361)(346,362)(347,364)(348,363)(349,381)(350,382)(351,384)(352,383)(353,377)(354,378)(355,380)(356,379)(357,373)(358,374)(359,376)(360,375), (1,61)(2,62)(3,63)(4,64)(5,67)(6,68)(7,65)(8,66)(9,70)(10,69)(11,72)(12,71)(13,49)(14,50)(15,51)(16,52)(17,55)(18,56)(19,53)(20,54)(21,58)(22,57)(23,60)(24,59)(25,85)(26,86)(27,87)(28,88)(29,91)(30,92)(31,89)(32,90)(33,94)(34,93)(35,96)(36,95)(37,73)(38,74)(39,75)(40,76)(41,79)(42,80)(43,77)(44,78)(45,82)(46,81)(47,84)(48,83)(97,157)(98,158)(99,159)(100,160)(101,163)(102,164)(103,161)(104,162)(105,166)(106,165)(107,168)(108,167)(109,145)(110,146)(111,147)(112,148)(113,151)(114,152)(115,149)(116,150)(117,154)(118,153)(119,156)(120,155)(121,181)(122,182)(123,183)(124,184)(125,187)(126,188)(127,185)(128,186)(129,190)(130,189)(131,192)(132,191)(133,169)(134,170)(135,171)(136,172)(137,175)(138,176)(139,173)(140,174)(141,178)(142,177)(143,180)(144,179)(193,253)(194,254)(195,255)(196,256)(197,259)(198,260)(199,257)(200,258)(201,262)(202,261)(203,264)(204,263)(205,241)(206,242)(207,243)(208,244)(209,247)(210,248)(211,245)(212,246)(213,250)(214,249)(215,252)(216,251)(217,277)(218,278)(219,279)(220,280)(221,283)(222,284)(223,281)(224,282)(225,286)(226,285)(227,288)(228,287)(229,265)(230,266)(231,267)(232,268)(233,271)(234,272)(235,269)(236,270)(237,274)(238,273)(239,276)(240,275)(289,349)(290,350)(291,351)(292,352)(293,355)(294,356)(295,353)(296,354)(297,358)(298,357)(299,360)(300,359)(301,337)(302,338)(303,339)(304,340)(305,343)(306,344)(307,341)(308,342)(309,346)(310,345)(311,348)(312,347)(313,373)(314,374)(315,375)(316,376)(317,379)(318,380)(319,377)(320,378)(321,382)(322,381)(323,384)(324,383)(325,361)(326,362)(327,363)(328,364)(329,367)(330,368)(331,365)(332,366)(333,370)(334,369)(335,372)(336,371), (1,85,37,49)(2,86,38,50)(3,88,39,52)(4,87,40,51)(5,96,44,58)(6,95,43,57)(7,93,42,59)(8,94,41,60)(9,90,47,55)(10,89,48,56)(11,91,45,54)(12,92,46,53)(13,73,25,61)(14,74,26,62)(15,76,27,64)(16,75,28,63)(17,84,32,70)(18,83,31,69)(19,81,30,71)(20,82,29,72)(21,78,35,67)(22,77,36,68)(23,79,33,66)(24,80,34,65)(97,181,133,145)(98,182,134,146)(99,184,135,148)(100,183,136,147)(101,192,140,154)(102,191,139,153)(103,189,138,155)(104,190,137,156)(105,186,143,151)(106,185,144,152)(107,187,141,150)(108,188,142,149)(109,169,121,157)(110,170,122,158)(111,172,123,160)(112,171,124,159)(113,180,128,166)(114,179,127,165)(115,177,126,167)(116,178,125,168)(117,174,131,163)(118,173,132,164)(119,175,129,162)(120,176,130,161)(193,277,229,241)(194,278,230,242)(195,280,231,244)(196,279,232,243)(197,288,236,250)(198,287,235,249)(199,285,234,251)(200,286,233,252)(201,282,239,247)(202,281,240,248)(203,283,237,246)(204,284,238,245)(205,265,217,253)(206,266,218,254)(207,268,219,256)(208,267,220,255)(209,276,224,262)(210,275,223,261)(211,273,222,263)(212,274,221,264)(213,270,227,259)(214,269,228,260)(215,271,225,258)(216,272,226,257)(289,373,325,337)(290,374,326,338)(291,376,327,340)(292,375,328,339)(293,384,332,346)(294,383,331,345)(295,381,330,347)(296,382,329,348)(297,378,335,343)(298,377,336,344)(299,379,333,342)(300,380,334,341)(301,361,313,349)(302,362,314,350)(303,364,315,352)(304,363,316,351)(305,372,320,358)(306,371,319,357)(307,369,318,359)(308,370,317,360)(309,366,323,355)(310,365,324,356)(311,367,321,354)(312,368,322,353), (1,289,97,193)(2,290,98,194)(3,291,99,195)(4,292,100,196)(5,293,101,197)(6,294,102,198)(7,295,103,199)(8,296,104,200)(9,297,105,201)(10,298,106,202)(11,299,107,203)(12,300,108,204)(13,301,109,205)(14,302,110,206)(15,303,111,207)(16,304,112,208)(17,305,113,209)(18,306,114,210)(19,307,115,211)(20,308,116,212)(21,309,117,213)(22,310,118,214)(23,311,119,215)(24,312,120,216)(25,313,121,217)(26,314,122,218)(27,315,123,219)(28,316,124,220)(29,317,125,221)(30,318,126,222)(31,319,127,223)(32,320,128,224)(33,321,129,225)(34,322,130,226)(35,323,131,227)(36,324,132,228)(37,325,133,229)(38,326,134,230)(39,327,135,231)(40,328,136,232)(41,329,137,233)(42,330,138,234)(43,331,139,235)(44,332,140,236)(45,333,141,237)(46,334,142,238)(47,335,143,239)(48,336,144,240)(49,337,145,241)(50,338,146,242)(51,339,147,243)(52,340,148,244)(53,341,149,245)(54,342,150,246)(55,343,151,247)(56,344,152,248)(57,345,153,249)(58,346,154,250)(59,347,155,251)(60,348,156,252)(61,349,157,253)(62,350,158,254)(63,351,159,255)(64,352,160,256)(65,353,161,257)(66,354,162,258)(67,355,163,259)(68,356,164,260)(69,357,165,261)(70,358,166,262)(71,359,167,263)(72,360,168,264)(73,361,169,265)(74,362,170,266)(75,363,171,267)(76,364,172,268)(77,365,173,269)(78,366,174,270)(79,367,175,271)(80,368,176,272)(81,369,177,273)(82,370,178,274)(83,371,179,275)(84,372,180,276)(85,373,181,277)(86,374,182,278)(87,375,183,279)(88,376,184,280)(89,377,185,281)(90,378,186,282)(91,379,187,283)(92,380,188,284)(93,381,189,285)(94,382,190,286)(95,383,191,287)(96,384,192,288)]
来源：https://stackoverflow.com/questions/48672107/sage-automorphism-group-of-finite-abelian-group